Resource Allocation for Single and Multiple Antenna OFDM ... · Resource Allocation for Single and Multiple Antenna OFDM Uplink and Downlink Systems von Diplom-Wirtschaftsingenieur

Resource Allocation for Single andMultiple Antenna OFDM Uplink and

Downlink Systems

von Diplom-Wirtschaftsingenieur

Thomas Michel

aus Darmstadt

von der Fakultat IV - Elektrotechnik und Informatik

der Technischen Universitat Berlin

zur Erlangung des akademischen Grades

Doktor der Ingenieurwissenschaften

- Dr.-Ing. -

genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. Dr. Bernd Rech

Berichter: Prof. Dr. Dr. Holger Boche

Berichter: Prof. Dr. Nihar Jindal (University of Minnesota)

Tag der wissenschaftlichen Aussprache: 24. April 2008

Berlin 2008

D 83

Zusammenfassung

Die vorliegende Arbeit beschaftigt sich mit der optimalenRessourcenvergabe in zellularen

Orthogonal Frequency Division Multiplexing (OFDM)-Mehrtragersystemen mit einer sowie

mehreren Antennen, wobei Aufwarts- und Abwartsstrecke behandelt werden. Besonderes In-

teresse gilt der Berucksichtigung von nutzerspezifischenRatenanforderungen, um der zuneh-

menden Bedeutung von Quality of Service in realen Systemen Rechnung zu tragen.

Im ersten Teil wird die optimale Leistungsallokation fur den Fall nur je einer Antenne

und zeit-invarianter Kanale untersucht. Drei Referenzprobleme werden vorgestellt und gelost,

wobei parallele Gauß-Kanale zur informationstheoretischen Modellierung von OFDM-System-

en dienen. Sowohl die Abwartsstecke als auch die Aufwartsstrecke liefern Moglichkeiten die

vorgestellten Referenzprobleme zu losen. Die Algorithmen, die die Formulierung der Aufwarts-

strecke nutzen, konnen hierbei besonders anschaulich alsWater-filling bezuglich allozierter

Raten interpretiert werden.

Basierend auf den erzielten Ergebnissen widmet sich der zweite Teil der verzogerungs-

beschranktenUbertragung mittels OFDM. Das Konzept der verzogerungsbeschrankten Ka-

pazitat bietet in diesem Kontext eine gute Moglichkeit das Potenzial verzogerungssensitiver

Nachrichtenubertragung zu quantifizieren. Um Bedingungen herzuleiten, unter denen ver-

zogerungsbeschrankteUbertragung moglich ist, werden Punkt-zu-Punkt-Verbindungen mit ei-

ner Antenne untersucht. Auch wird in diesem Rahmen der Einfluß verschiedener Systempa-

rameter auf die verzogerungsbeschrankte Kapazitat charakterisiert. Der folgende Teil wendet

sich der komplizierteren Abwartsstrecke zu, deren verzogerungsbeschrankte Kapazitatsregion

man mit Hilfe eines iterativen Algorithmus evaluieren kann. Ferner werden untere Schranken

fur diese Region prasentiert. Der letzte Abschnitt schließlich untersucht das fur die Praxis

sehr wichtige Problem der Leistungsminimierung bei gegebenen Ratenanforderungen, wenn

zusatzlich zu berucksichtigen ist, dass Untertrager annur je einen einzigen Nutzer vergeben

werden durfen. Die prasentierten Algorithmen zur Losung des Problems konnen als duale Op-

timierungsverfahren interpretert werden und verallgemeinern Ideen aus dem ersten Teil der Ar-

beit.

Der dritte Teil erweitert die Untersuchung optimaler Leistungsallokationen fur zeit-inva-

riante Kanale auf den Fall mehrerer Antennen. Anders als imvorherigen Fall mit nur einer An-

tenne ist eine Losung der Referenzprobleme fur die Abwartstrecke nicht mehr direkt moglich.

Es mussen Dualitatsbeziehungen genutzt werden um die Probleme uber den Umweg der Auf-

i

ii Zusammenfassung

wartsstrecke anzugehen. Der Grund hierfur ist in der mathematischen Struktur zu finden,

welche die Existenz raumlicher Freiheitsgrade im Fall mehrerer Antennen stark verandert.

Im weiteren Verlauf wird der Einfluss der Nutzerzahl auf die Summenrate charakterisiert und

fur den Fall einer Summenratenanforderung eine algorithmische Losung prasentiert, die ein

sehr gutes Konvergenzverhalten auch fur Probleme mit vielen Nutzern und vielen Untertragern

aufweist. Abschließend werden Moglichkeiten diskutiertdie dualen Optimierungsansatze im

Fall exklusiver Untertragervergabe fur Szenarien mit mehreren Antennen zu verallgemeinern.

Abstract

In this thesis the optimum allocation of resources in singleand multiple antenna Orthogonal

Frequency Division Multiplexing (OFDM) uplink and downlink systems is studied. A special

focus lies on the incorporation of minimum rate constraintsin order to reflect the increasing

importance of Quality of Service requirements in real systems.

In the first part we investigate the optimum power allocationfor time-invariant channels in

the single antenna case. To this end we solve three referenceproblems modeling OFDM systems

as a set of parallel Gaussian channels. We present a downlinkapproach as well as an equivalent

uplink formulation where the derived uplink algorithms have an appealing interpretation as

water-filling with respect to rates.

Equipped with these results we turn towards the topic of delay-limited transmission over

single antenna OFDM channels in the second part. The delay-limited capacity is an important

metric to quantify the possibilities for delay-sensitive communication. As a first step, point-to-

point links are studied in order to derive conditions on the existence of a non-zero delay-limited

capacity and to understand the general dependency on systemparameters. Subsequently we

turn towards the more complicated downlink and derive an algorithm to evaluate the OFDM

delay limited capacity region as well as lower bounds. The last section of this part focuses

on the practically important topic of power minimization under rate constraints with exclusive

subcarrier assignment and presents algorithms generalizing principles from the first part.

Thereafter we extend our study of optimum resource allocation for time-invariant channels

to the case of multiple antennas in part three. We study the introduced reference problems in

the uplink and use recent duality results to transform the solutions to the dual downlink. A

direct downlink formulation is not possible anymore since the necessary mathematical structure

is lost due to the additional spatial degrees of freedom. Taking the fading statistics into account

we then characterize the influence of the number of users on the multiple antenna OFDM sum

capacity and on the sum rate achievable with exclusive subcarrier assignment. Further, we

derive a rate water-filling algorithm for the case of a throughput constraint which exhibits an

excellent convergence behavior. Finally, we comment on thepossibilities to generalize the dual

optimization methods used for single antenna systems underexclusive subcarrier assignment

constraints to scenarios with multiple antennas.

iii

Acknowledgment

During my time at the Fraunhofer German-Sino Lab for Mobile Communications I came across

excellent conditions and an extraordinary stimulating atmosphere. This was due to the never

ending enthusiasm of my advisor Professor Holger Boche who Iam indebted to. I also thank

Professor Nihar Jindal very much for dedicating his time andenergy to act as second referee of

this thesis. He not only supported me with valuable advice but also made possible my stay at

the University of Minnesota. Further I would like to expressmy special gratitude to Gerhard

Wunder and my room mate Zhou Chan for sharing countless ideasand discussions. Without

them the last years would not have been the same. My gratitudeextends to all colleagues of the

Fraunhofer German-Sino Lab for Mobile Communications, theHeinrich-Hertz-Institut and the

Technical University of Berlin.

Moreover, I would like to thank my parents and my grandfatherWerner for their unquestioning

support. Last but not least my deepest thanks goes to my girlfriend Marion who had to put up

with me all the time.

v

Contents

1 Introduction 1

1.1 Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Information-Theoretic Modeling of OFDM Uplink and Downli nk 5

2.1 Orthogonal Frequency Division Multiplexing. . . . . . . . . . . . . . . . . . 5

2.1.1 Communication Link Model. . . . . . . . . . . . . . . . . . . . . . . 6

2.1.2 Cyclic Prefixed OFDM. . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 System Model of Single Antenna OFDM BC and MAC. . . . . . . . . . . . . 8

2.2.1 Scalar Gaussian Broadcast Channel. . . . . . . . . . . . . . . . . . . 9

2.2.2 Scalar Gaussian Multiple-Access Channel. . . . . . . . . . . . . . . . 10

2.2.3 Duality of Scalar Gaussian BC and MAC. . . . . . . . . . . . . . . . 12

2.2.4 Parallel Scalar Gaussian BCs and MACs. . . . . . . . . . . . . . . . . 13

2.3 System Model of Multiple-Antenna OFDM BC and MAC. . . . . . . . . . . 14

2.3.1 MIMO Gaussian Broadcast Channel. . . . . . . . . . . . . . . . . . . 15

2.3.2 MIMO Gaussian Multiple-Access Channel. . . . . . . . . . . . . . . 17

2.3.3 Duality of MIMO Gaussian BC and MAC. . . . . . . . . . . . . . . . 18

2.3.4 Parallel MIMO Gaussian BCs and MACs. . . . . . . . . . . . . . . . 19

2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Resource Allocation for Single Antenna OFDM Systems 23

3.1 Problem Statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2 A Broadcast Channel Approach. . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2.1 Characterization of the Capacity Region in the BC Domain . . . . . . . 27

3.2.2 Maximizing a Weighted Sum of Rates. . . . . . . . . . . . . . . . . . 29

3.2.3 Sum Power Minimization. . . . . . . . . . . . . . . . . . . . . . . . 31

3.2.4 Introducing Minimum Rates. . . . . . . . . . . . . . . . . . . . . . . 32

3.3 A Multiple-Access Channel Approach. . . . . . . . . . . . . . . . . . . . . . 36

3.3.1 Weighted Sum Power Minimization. . . . . . . . . . . . . . . . . . . 37

3.3.2 Extension to Minimum Rate Requirements. . . . . . . . . . . . . . . 42

3.4 Individual vs. Global Successive Decoding Order. . . . . . . . . . . . . . . . 44

vii

viii CONTENTS

3.4.1 Optimum Decoding Order for Parallel MACs. . . . . . . . . . . . . . 44

3.4.2 Optimum Decoding Order for Parallel BCs. . . . . . . . . . . . . . . 45

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4 Delay-Limited Transmission over OFDM Broadcast Channels 49

4.1 Single User OFDM Delay-Limited Capacity. . . . . . . . . . . . . . . . . . . 50

4.1.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.1.2 Rate Water-Filling and Single User OFDM DLC. . . . . . . . . . . . 51

4.1.3 Low SNR Behavior. . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.1.4 High SNR Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2 Multi-User OFDM Delay-Limited Capacity. . . . . . . . . . . . . . . . . . . 63

4.2.1 The OFDM BC Delay-Limited Capacity Region. . . . . . . . . . . . 63

4.2.2 An Algorithm to Evaluate the OFDM BC DLC Region. . . . . . . . . 64

4.3 Delay-Limited Transmission under FDMA Constraints. . . . . . . . . . . . . 65

4.3.1 Lower Bounds on the OFDM BC DLC Region Based on FDMA. . . . 66

4.3.2 Conditions for Optimality of FDMA Transmission. . . . . . . . . . . 72

4.3.3 Sum Power Minimization under FDMA Constraints. . . . . . . . . . 74

4.3.4 Performance Bounds Via Duality Gap. . . . . . . . . . . . . . . . . . 81

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5 Resource Allocation for Multiple-Antenna OFDM Systems 85

5.1 Weighted Rate-Sum and Throughput Maximization. . . . . . . . . . . . . . . 86

5.1.1 Weighted Rate-Sum Maximization. . . . . . . . . . . . . . . . . . . . 86

5.1.2 Algorithmic Approaches. . . . . . . . . . . . . . . . . . . . . . . . . 88

5.1.3 Optimality Conditions for FDMA Transmission. . . . . . . . . . . . . 89

5.1.4 Influence of the Number of Users on the MIMO-OFDM sum capacity . 91

5.2 Power Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.2.1 Power Minimization under Individual Rate Constraints . . . . . . . . . 100

5.2.2 Sum Rate Iterative Water-Filling. . . . . . . . . . . . . . . . . . . . . 101

5.3 Minimum Rate Requirements. . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.4 Comments on FDMA Constrained Transmission. . . . . . . . . . . . . . . . . 123

5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6 Conclusions 127

Appendix 129

Publication List 135

Bibliography 144

List of Figures

1.1 Downlink and uplink of a cellular communication system. . . . . . . . . . . . 2

2.1 Gaussian broadcast channel and its dual multiple-access channel . . . . . . . . 9

2.2 Exemplary polymatroid multiple-access channel rate region . . . . . . . . . . . 11

2.3 Gaussian MIMO broadcast channel and its dual MIMO multiple-access channel 15

3.1 Illustration of Problem 1: Weighted rate-sum maximization . . . . . . . . . . . 24

3.2 Illustration of Problem 2: Sum power minimization. . . . . . . . . . . . . . . 26

3.3 Illustration of Problem 3: Weighted rate-sum maximization with rate constraints 27

3.4 Enhanced setG(g) and exemplary capacity regionC(g, P) . . . . . . . . . . . . 31

3.5 Convergence behavior of broadcast channel minimum rates algorithm . . . . . 36

3.6 Exemplary power regionP(g, R) . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.7 Convergence behavior of sum power minimization algorithm . . . . . . . . . . 42

3.8 Illustration of the optimum decoding order in presence of minimum rates . . . 46

3.9 RatesR and Lagrangian multipliersµ over SNR for minimum rates algorithm. 47

4.1 1st and 2nd order behavior of the OFDM delay-limited capacity overEb/N0 . . . 59

4.2 OFDM DLC overEb/N0 at low SNR for different number of taps. . . . . . . . 59

4.3 OFDM delay-limited capacity and upper bound at high SNR. . . . . . . . . . 63

4.4 Illustration of OFDM delay-limited capacity region evaluation . . . . . . . . . 65

4.5 OFDM BC DLC region for 16 subcarriers and lower bounds at -20 dB . . . . . 70

4.6 OFDM BC DLC region for 16 subcarriers and lower bounds at 10 dB . . . . . 71

4.7 Comparison of FDMA algorithm with optimum transmissionscheme. . . . . . 80

5.1 Different fixed step-sizes for sum rate iterative water-filling. . . . . . . . . . . 112

5.2 Optimization of step-sizeν for sum rate iterative water-filling. . . . . . . . . . 113

5.3 Comparison of convergence speed for 10 and 100 users. . . . . . . . . . . . . 114

5.4 Exemplary optimum polymatroid for sum rate iterative water-filling . . . . . . 115

5.5 Illustration of ellipsoid minimum rates algorithm. . . . . . . . . . . . . . . . 120

5.6 Exemplary convergence of ellipsoid minimum rates algorithm . . . . . . . . . 121

5.7 RatesR and multipliersµ over SNR for MIMO minimum rates algorithm. . . 123

ix

Chapter 1

Introduction

1.1 Motivation

The core of most wireless communication systems today is andsupposedly for the foreseeable

future will be a cellular structure. This means that the areacovered by a system is subdivided

into smaller units, so-called cells, each served by a base station. The base stations of different

cells are wired and interconnected. Every communication link between a pair of users is estab-

lished via the base stations of the corresponding cells the users are assigned to. The wireless

communication link between the base station and multiple users within the cell is known as

downlink, the opposite direction between multiple users and the base station is called uplink.

Both links will be termed multi-user channels in the following and constitute the central in-

stances of any cellular scheme. They are illustrated in Figure 1.1. A deep understanding of

these two multi-user channels is an indispensable prerequisite to effectively engineer cellular

communication systems.

Concerning the signal processing and transmission technology in cellular systems, Time Di-

vision Multiple Access (TDMA) and Frequency Division Multiple Access (FDMA) have been

the predominant access schemes for a long time. Most second generation mobile telecommu-

nications standards including the Global System for MobileCommunications (GSM, Groupe

Speciale Mobile) are still based on these simple orthogonalaccess techniques. Code Divi-

sion Multiple Access (CDMA) emerged and complemented the former two access technolo-

gies as it entered the Interim Standard 95 (IS-95) and later the CDMA2000 protocol. CDMA

has the considerable advantage to be much more robust against narrow-band noise distur-

bances due to the spread spectrum properties. It became the undisputed leading technology

and found entrance in the European third generation Universal Mobile Telecommunications

System (UMTS)-standard.

Parallel the above-described developments, the multicarrier scheme Orthogonal Frequency

Division Multiplexing (OFDM) gained importance in the areaof (fixed) broadband communi-

cations, because it can be very efficiently implemented by means of the Fast Fourier Transform

1

2 Chapter 1. Introduction

Figure 1.1: Downlink (left) and uplink (right) of a cellularcommunication system

(FFT) simplifying the signal processing significantly. Further, OFDM decouples the frequency

selective channel into a set of parallel frequency flat subcarriers so that channel equalization be-

comes fairly simple. Due to these advantages OFDM has becomealready a fundamental part of

the third generation long term evolution (LTE) [24] and is currently creeping into further stan-

dards. Independent of the former development, the application of multiple antennas at either

the base station or even both sides of the wireless link became a new paradigm after Telatar [25]

and Foschini together with Gans [26] revealed the significant capacity gains of multiple input

multiple output (MIMO) systems. Both trends have been evolving independently but comple-

ment one another very well: The synergetic combination of OFDM and MIMO is promising

and on its way to becoming the base of systems beyond the thirdgeneration. In fact, some of the

latest finalized standards such as e.g. WiMAX 802.16e [27] and WLAN 802.11n [28] already

allow for a limited combined application of both OFDM and multiple antennas.

However, MIMO-OFDM is still in the fledgling stages and recent demonstrator implemen-

tations showed that many problems remain unsolved. But not only practical implementations

create questions: Multi-user MIMO-OFDM systems constitute a challenge also from a theo-

retic point of view. While power in single antenna OFDM systems can be allocated over the

two dimensions time and frequency, multiple antenna systems have to integrate the additional

spatial dimension. Further, the coding concepts suited forsingle antenna multi-user channels

fail. These facts complicate the analysis of multi-user MIMO-OFDM systems significantly,

making resource allocation a complex task. But it is not onlymultiple antenna systems which

offer challenges. Even basic questions concerning the optimalallocation of resources in OFDM

multi-user systems with a single antenna remain unsolved. The solutions to these problems are

not only interesting from a theoretic perspective: They also allow for structural insights in the

behavior of MIMO-OFDM systems. This means they yield guidelines how to allocate the scarce

resources power and bandwidth effectively also under real-world conditions, thus providinga

1.2. Outline 3

valuable contribution to the engineering of future wireless communication systems.

This thesis is devoted to the study of resource allocation problems for single antenna and

multiple antenna OFDM multi-user channels adopting a fundamental information-theoretic

point of view. A main focus lies on the incorporation of user specific individual rate con-

straints, which can be used to model Quality of Service (QoS)-requirements on a physical

layer-level. With the current evolution of sophisticated delay-sensitive services such as for

instance video/audio live streaming or other real-time multimedia applications, the question of

finding transmit strategies, which allow guaranteed rates,has become crucial. We begin with a

study of single antenna systems and time-invariant channels. We show connections between the

different problem formulations and their solutions and presentcorresponding algorithms. Mo-

tivated by the need for guaranteeable rates we then leave thedomain of time-invariant channels

behind and carry out an analysis for the recently introducedmetric of delay limited capacity.

Finally we return to the case of time-invariant channels andextend our study to scenarios with

multiple antennas.

1.2 Outline

The remainder of this thesis is organized as follows.

Chapter 2 introduces the system model and the information-theoreticnotions of parallel

Gaussian broadcast and multiple-access channels. They areappropriate to model single- and

multiple antenna multi-user OFDM systems and serve as the fundamental concepts through-

out this thesis. The chapter begins with a brief review of OFDM as a transmission technique

decomposing any time-invariant channel into a set of parallel channels in Section2.1. Single

antenna OFDM broadcast and multiple-access channels as well as basic duality results between

uplink and downlink are introduced in Section2.2. Section2.3generalizes the system model to

the case of multiple antennas.

In Chapter 3a framework for optimum resource allocation in single antenna OFDM up- and

downlink systems with fixed channel gains is developed. Three reference problems are intro-

duced in Section3.1, allowing for minimum rate constraints which represent QoS-requirements

on a physical layer level. The introduced problems are solved for the downlink scenario in Sec-

tion 3.2and for the uplink scenario in Section3.3. The chapter concludes with a discussion of

the optimal successive decoding orders for both cases.

Chapter 4 studies the metric of delay-limited capacity for OFDM broadcast channels. Sec-

tion 4.1focuses on the single user OFDM delay limited capacity, while the multi-user downlink

case follows in Section4.2. Since in general the optimum transmit strategy involves superposi-

tion coding and thus is very complex, we investigate simple schemes with exclusive subcarrier

allocation, i.e. orthogonal signaling based on FDMA in Section 4.3. Conditions for the opti-

mality of FDMA transmission are derived. An algorithm for finding close-to-optimal FDMA

strategies is presented and a general bound on the performance loss for suboptimal strategies is

4 Chapter 1. Introduction

derived.

Chapter 5 considers the multi-antenna case. We carry out an analysis of the three reference

problems introduced in the context of single antenna systems in Sections5.1, 5.2and5.3. The

algorithms developed for the single antenna case do not apply here and different algorithmic

solutions are found. Moreover, conditions for the optimality of FDMA are derived and we

comment on the generalization of FDMA-algorithms to multiple-antenna settings.

Finally, Chapter 6 draws conclusions.

The appendix contains basic definitions and results about polymatroids and contra-poly-

matroids needed throughout this thesis.

1.3 Notations

Upper case and lower case bold letters represent matrices and column vectors, respectively. The

superscript (·)T denotes the transpose of a vector or matrix and (·)H the conjugate transpose. Sets

are denoted by calligraphic letters. The trace operator is given by tr(·), and the determinant will

be denoted as| · |. X � Y means thatX − Y is a positive semidefinite matrix andX ≻ Y says

thatX − Y is a positive definite matrix. We useλmax(·) andλmin(·) to denote the maximum and

minimum eigenvalue of a normal matrix. log(·) is the natural logarithm,E{·} is the expectation

operator for a random variable and Pr{·} denotes the probability of an event. The operator

[·]+ is equivalent to max(·, 0). The imaginary unit is given byj. The set of real and complex

numbers is denoted byR andC, respectively. We say that a random variablea = b + jc

is complex Gaussian distributeda ∼ CN(0, 1) if its real and imaginary part are independently

Gaussian distributed withb ∼ N(0, 1/2) andc ∼ N(0, 1/2). Further we say that a random vector

c ∈ CN is circular symmetric complex Gaussian distributed withc ∼ CN(0N,R) if its entries

are identically Gaussian distributed with meanE{c} = 0N and covariance matrixE{ccH} = R,

where0N is the all-zero vector of lengthN. Similarly we use1N to denote the all-one vector

andI N for theN × N identity matrix.

Chapter 2

Information-Theoretic Modeling of OFDM

Uplink and Downlink

This chapter introduces parallel Gaussian single and multiple antenna broadcast channels (BC)

and multiple-access channels (MAC). The motivation is thatfrom a theoretic point of view, the

downlink of a cellular OFDM system can be modeled as a set of parallel Gaussian BCs and the

uplink as a set of parallel Gaussian MACs. This is due to the fact that OFDM decouples the

time-invariant frequency selective channel into a set of orthogonal channels. Most likely the

first who considered parallel BCs was Hughes-Hartogs [29]. In 1975 he derived the capacity

region for parallel degraded Gaussian BCs and gave a rate region for general parallel Gaussian

BCs. The first occurrence of parallel MACs dates back to the seventies of the past century

[30]. The concept ofdegradednessis crucial in the study of systems with a single antenna.

Degradedness means that the channel can be represented by a Markov-chain, simplifying the

analysis significantly [31]. Unfortunately this concept does not carry over to multiple antenna

systems and thus it was not until recently that the MIMO BC wascompletely understood [32,

33, 34, 35, 36]. The MIMO MAC was studied already before in [37] in the context of single

antenna MACs with intersymbol-interference. For both the single and the multiple antenna case

a duality relationship exists connecting the broadcast- tothe multiple-access scenario [38, 34].

These relationships are essential throughout this thesis as they allow to connect results from the

MAC to the BC and vice versa.

We begin with a short review of OFDM as the basic modulation technique in Section2.1.

Subsequently, scalar parallel Gaussian BCs and MACs are introduced in Section2.2 while

Section2.3extends the system model to multiple antennas.

2.1 Orthogonal Frequency Division Multiplexing

This section is intended to briefly introduce OFDM being one of the standard modulation tech-

niques in modern communication systems. Its history dates back to the late sixties of the past

5

6 Chapter 2. Information-Theoretic Modeling of OFDM Uplink and Downlink

century, where in the context of discrete multi-tone transmission the basic ideas of OFDM were

developed [39, 40]. The key property of OFDM which is of particular importanceis that it

decomposes a static frequency selective channel into a set of parallel Gaussian channels. For a

deeper study especially of the signal processing details and impairments the reader is referred

to one of the numerous existing text books, such as e.g. [41].

2.1.1 Communication Link Model

Consider a communication link over a time-discrete time-invariant frequency selective channel.

Let the sampled baseband channel impulse response be given by a sequence of equidistant com-

plex channel gainscl ∈ C, l = 1, ..., L each corresponding to a resolvable path. The maximum

number of nonzero samples is denoted byL and thus the length of the channel impulse response

is (L−1)Ts, wherefs = 1/Ts is the sampling frequency∗. Assume further that the channel output

is corrupted by an additive white Gaussian thermal noise process. Then the communication link

can be represented by the linear time-invariant system

yt =

L−1∑

τ=0

dt−τcτ+1 + nt, (2.1)

wheredt ∈ C denotes the channel input,yt ∈ C the received signal andnt ∼ CN(0, σ2n) is an

i.i.d. additive Gaussian noise process, each at time instant t. Equation (2.1) is the basic system

equation of any discrete time-invariant communication link.

2.1.2 Cyclic Prefixed OFDM

We wish to transmit a data sequence

d = [d1, ..., dK]T (2.2)

over the communication link introduced in (2.1). For reasons to become clear in the following

a cyclically extended version of the data sequence is sent sothat theL − 1 last samples precede

the original sequence:

dCP = [dK−L+1, ...dK, d1, d2, ..., dK]T . (2.3)

Moving to matrix notation and removing the firstL − 1 received samples the vector of received

samplesy ∈ CK can be written as

y = Cd + n (2.4)

∗For notational convenience the index corresponding to zerodelay is chosen to be 1.

2.1. Orthogonal Frequency Division Multiplexing 7

wheren ∈ CK is an additive noise vector with i.i.d. entriesnt ∼ CN(0, σ2n) for t = 1, ...,K and

the channel matrix is given by

C =

c1 0 . . . 0 cL . . . c2

c2 c1 0 . . . 0 cL . . . c3

. . .

cL . . . c2 c1 0 . . . 0

0 cL . . . c2 c1 0 . . . 0. . .

0 . . . 0 cL . . . c2 c1

. (2.5)

The upper right corner of (2.5) reflects the echoes of the prefixed samples. Note that (2.4) has

the form of a cyclic convolution of (2.2) and (2.5) is acirculant matrixyielding the key property

of OFDM. Since circulant matrices belong to the class of normal matrices they are unitarily

equivalent to a diagonal matrix. In fact the Discrete Fourier Transform (DFT) diagonalizes any

circulant matrix

C =WHΛW, (2.6)

where

W = [wm,n], wm,ndef=

√

1K

exp

(

− j2π(m− 1)(n− 1)K

)

is the DFT matrix andΛ ∈ CK×K is a diagonal matrix containing the eigenvalues ofC. Substi-

tuting (2.6) into (2.4) and multiplying byW from the left hand side leads to

Wy =WW HΛWd +Wn.

Introducing variablesy =Wy, d =Wd andn =Wn yields

y = Λd + n, (2.7)

where fromn ∼ CN(0, σ2nI ) it follows that the resulting noise has the same isotropic distribution

n ∼ CN(0, σ2nI ). As a consequence (2.7) decomposes to a set ofK orthogonal channels

yk = λk,kdk + nk k = 1, ...,K

whereλk,k is thekth diagonal element ofΛ. These orthogonal channels form thesubcarriersin

an OFDM system. To summarize, performing an inverse DFT at the transmitter and a DFT at

the receiver side orthogonalizes the system and thus simplifies equalization significantly. The

physical interpretation behind this principle is that transmission along the eigenvectors ofC,

i.e. the rows ofW, decouples the frequency selective channel toa set of K parallel Gaussian

channels.

8 Chapter 2. Information-Theoretic Modeling of OFDM Uplink and Downlink

2.2 System Model of Single Antenna OFDM Broadcast and

Multiple-Access Channel

In this section we introduce the basic single antenna OFDM system model. The case of multiple

antennas follows subsequently in Section2.3.

Consider a time discrete time-invariant frequency selective multi-user channel over which a

base station andM users, all equipped with a single antenna, communicate witheach other. It

is assumed that all users as well as the base station know the time-invariant channel realization

perfectly. As described in the previous section the application of OFDM turns the constant fre-

quency selective point-to-point channel into a set ofK parallel Gaussian channels, one on each

subcarrier. Assuming perfect synchronization the same obviously holds for a time-invariant

frequency selective multi-user channel: Letcm = [cm,1, ..., cm,L]T be the time-invariant channel

impulse response of userm. Applying OFDM its channel transfer function is given by means

of the FFT as

hm,k =

L∑

l=1

cm,l exp

(

− j2π(l − 1)(k− 1)K

)

m ∈ M, k ∈ K (2.8)

where we useM = {1, ...,M} andK = {1, ...,K} to denote the set of users and subcarriers,

respectively. The system equation on subcarrierk ∈ K of the OFDM downlink is then given by

yDLm,k = hm,kxDL

k + nDLm,k ∀ m ∈ M (2.9)

whereyDLm,k ∈ C is the signal received by usermon subcarrierk, xDL

k ∈ C is the signal transmitted

on subcarrierk andnDLm,k ∼ CN(0, 1) is additive white Gaussian thermal noise with normalized

variance. The case of arbitrary noise variances can be included simply by scaling the channel so

that there is no loss of generality in this assumption. The corresponding uplink system equation

with the same set of channel realizations reads as

yULk =

M∑

m=1

hm,kxULm,k + nUL

k (2.10)

whereyULk ∈ C is the signal received by the base station on subcarrierk, xUL

m,k ∈ C denotes the

signal transmitted by userm on subcarrierk andnULk ∼ CN(0, 1) again is normalized additive

white Gaussian receiver noise. Eqn. (2.9) represents a scalar Gaussian broadcast channel and

eqn. (2.10) a scalar multiple-access channel. Both channels are depicted in Figure2.1 where

we dropped the superscripts·DL and·UL for the ease of notation.

In order to study a set ofK parallel scalar BCs and MACs as it is necessary for OFDM we

focus on the elementary BC and MAC first.

2.2. System Model of Single Antenna OFDM BC and MAC 9

+

+

+

+xk

x1,k

x2,k

xM,k

yk

y1,k

y2,k

yM,k

h1,k

h2,k

hM,k

h1,k

h2,k

hM,k

nk

n1,k

n2,k

nM,k

Figure 2.1: Gaussian broadcast channel (left) and its dual multiple-access channel (right) onsubcarrierk

2.2.1 Scalar Gaussian Broadcast Channel

For the purpose of a concise notation the channel indexk is omitted as long as we deal with an

isolated BC. We pick up the carrier indexk once again as we return to parallel channels.

Consider one of the parallel Gaussian BCs. The signal received by userm is given by

ym = hmx+ nm ∀ m ∈ M (2.11)

wherehm ∈ C is the channel between the transmitter and userm, x ∈ C is the transmit signal and

nm ∼ CN(0, 1) is additive white Gaussian noise with unit variance. Assume that the transmit

power is limited to

E{

|x|2}

≤ P.

The elementary BC in (2.11) is adegradedBC. A degraded broadcast channel can be rep-

resented by a Markov chain

x→ yπ(1) → · · · → yπ(M).

This means that there exists a permutationπ(·) : M 7→ M and a set of conditional probability

density functionsp(yπ(1)|x), p(yπ(m)|yπ(m−1)) for m≥ 2 so that

p(y1, ..., yM|x) = p(yπ(1)|x)M∏

m=2

p(yπ(m)|yπ(m−1)),

i.e. each receiver sees a degraded version of the signal the previous user received [31]. The

capacity region of the degraded broadcast channel was derived in [42]†. For the special case

of a Gaussian BC the capacity region under a sum power constraint P is well established [31,

Sec. 14.6]. Introducing the channel gainsgm = |hm|2 andg = [g1, ..., gM]T the set of rate tuples

†The capacity region of a general BC remains still open.

10 Chapter 2. Information-Theoretic Modeling of OFDM Uplin k and Downlink

achievable with powerP can be expressed as

CBC

(

g, P)

=⋃

∑Mm=1 pm=P

{

R : Rπ(m) ≤ log

(

1+gπ(m)pπ(m)

1+ gπ(m)∑

n<m pπ(n)

)

∀ m ∈ M}

(2.12)

whereπ(·) is the introduced permutation defining the decoding order such that

π(·) : gπ(1) ≥ ... ≥ gπ(M). (2.13)

Note that the phases of the channel realizations are irrelevant and the capacity region only

depends on the absolute values, i.e. the channelgainsg. We thus speak of channel realizations

and channel gain realizations synonymously for the single antenna case in the following.

Each rate tuple inside the capacity region can be achieved using Superposition Coding and

successive decoding. To be more precise, the transmitter chooses from its codebook a codeword

consisting of superimposed Gaussian codewords for allM users and each userm successively

decodes all codewords of usersn with weaker channel gain, i.e. for alln with gn < gm before

decoding his own codeword. With (2.13) given userπ(M) is decoded first, followed by user

π(M − 1) and so on. This is the traditional way to look at degraded Gaussian BCs. As shown in

[43] the capacity region can also be achieved with a more sophisticated coding scheme based on

interference pre-compensation and the ideas in [44]. This coding scheme namedDirty-Paper

Codinghowever is more general and will prove useful in Section2.3.1 in the case of multi-

antenna Gaussian BCs which are not degraded any more.

2.2.2 Scalar Gaussian Multiple-Access Channel

Now we consider a MAC with the same set of channelsh1, ..., hM as the BC in the previous

section. The right hand side of Figure2.1 depicts one of theK elementary MACs. Dropping

the subcarrier indexk the signal received by the receiver is given by a superposition of theM

transmitted signals

y =M∑

m=1

hmxm+ n (2.14)

wherexm ∈ C is the signal transmitted by userm andn ∼ CN(0, 1) is additive white Gaussian

noise with unit variance. The power of each userm is limited to a budgetE{

|xm|2}

≤ Pm.

The characterization of the capacity region of this scalar MAC makes use of some definitions

and results from the appendix. The capacity region of this scalar MAC is given by [31]

CMAC

(

g, P1, .., PM

)

=

R :

∑

m∈IRm ≤ log

1+∑

m∈IgmPm

∀ I ⊆ M

. (2.15)

Rate tuples on the boundary are achieved if each user exhausts its power budget. It can be easily


R1

R2

R3

Figure 2.2: Exemplary polymatroid multiple-access channel rate region forM = 3 users

verified that

f (I)def= log

1+∑

m∈IgmPm

, I ⊆ M (2.16)

is a submodular rank function according to DefinitionA.1 in the appendix. With Definition

A.2 it becomes obvious that the rate region given in (2.15) is a polymatroid. An exemplary

rate region with polymatroid structure is depicted in Figure 2.2 for the case of three users. The

equivalent DefinitionA.3 of a polymatroid states that there areM! vertices of a polymatroid

lying strictly inside the positive orthant each corresponding to a different permutationπ(·) on

the setM. As a consequence each vertex of (2.15) corresponds to one of theM! possible

successive decoding orders and according to (A.2) the achievable rate tuples are given by

Rπ(m) = log

1+m∑

n=1

gπ(n)pπ(n)

− log

1+m−1∑

n=1

gπ(n)pπ(n)

= log

(

1+gπ(m)pπ(m)

1+∑m−1

n=1 gπ(n)pπ(n)

)

∀ m ∈ M, (2.17)

where the decoding orderπ(·) is such that userπ(M) is decoded first, followed by userπ(M −1) and so on. Coming back to Figure2.2 there are 6 different vertices corresponding to the

decoding orders{1, 2, 3}, {1, 3, 2}, {2,1, 3}, {2, 3, 1}, {3, 1,2} and{3, 2, 1}.

If in contrast a sum power constraint over all usersP =∑M

m=1 Pm is assumed the rate region

consists of an uncountable union of polymatroids where eachcorresponds to a specific power

allocation. In this case there exists one decoding order achieving all points on the boundary of

the capacity region. The decoding order is the reverse of that in the scalar BC given in (2.13)


using Superposition Coding

π(·) : gπ(M) ≥ ... ≥ gπ(1). (2.18)

so that userπ(M) is decoded first, followed by userπ(M − 1) and so on.

2.2.3 Duality of Scalar Gaussian BC and MAC

An important result relating the scalar BC to its dual MAC waspresented by Jindal et al. in

[38].

Theorem 2.1. [38, Thm. 1] Given channel gainsg ∈ RM+ the following identity holds with

the capacity region of the Gaussian BC given in(2.12) and the Gaussian MAC capacity region

given in(2.15):

CBC

(

g, PBC)

=⋃

∑Mm=1 PMAC

m =PBC

CMAC

(

g, PMAC1 , ..., PMAC

M

)

. (2.19)

In other words the capacity region of the Gaussian BC under a power constraintPBC is

equal to the union of capacity regions of the dual MAC with theset of power constraints

PMAC1 , ..., PMAC

M such that∑M

m=1 PMACm = PBC.

As a result each rate tuple achievable in the BC can be achieved in the dual MAC as well

if the same sum power is available. Nevertheless the power allocation achieving a certain rate

tuple in the BC differs from the power allocation achieving the same rate tuple in the MAC. The

transformations between these power allocations were alsoderived in [38]. The MAC power

allocationpMAC1 , ..., pMAC

M achieving the same rate tuple as the BC power allocationpBCπ(1), ..., p

BCπ(M)

is given by the recursive expression

pMACπ(m) = pBC

π(m)

1+∑M

n=m+1 gπ(n)pMACπ(n)

1+ gπ(m)∑m−1

n=1 pBCπ(n)

, (2.20)

where the decoding order for the BC is defined in (2.13). Conversely the BC power allocation

pBC1 , ..., pBC

M achieving the same rate tuple as a certain MAC power allocation pMACπ(1) , ..., p

MACπ(M) can

be calculated as

pBCπ(m) = pMAC

π(m)

1+ gπ(m)∑m−1

n=1 pBCπ(n)

1+∑M

n=m+1 gπ(n)pMACπ(n)

, (2.21)

where the decoding order for the dual MAC is specified by (2.18). Note that the ordering has

to be reversed. These transformations will prove useful since they allow to solve downlink

problems in the MAC. The optimum power allocations then can be simply transformed to the

corresponding BC scenario.


2.2.4 Parallel Scalar Gaussian BCs and MACs

With the results from the previous two subsections at hand wecan address the case of parallel

channels. The BC in (2.9) consists ofK parallel scalar BCs. Although each of the component

channels is degraded, the overall system is anon-degradedbroadcast channel. In principle it is

not possible to achieve the capacity region of an arbitrary non-degraded BC with Superposition

Coding. However, in the special case of parallel degraded BCs the capacity region of the entire

system under a sum power constraintP =∑K

k=1 Pk with Pk being the power allocated to thekth

BC can still be achieved using Superposition Coding and successive decoding. To this end the

transmitter uses a different codeword on each of theK parallel BCs and for eachk the receivers

apply anindividual successive decoding orderπk(·) according to the ordering of the channel

gains on each channelk:

πk(·) : gπk(1),k ≥ ... ≥ gπk(M),k. (2.22)

Picking up the subcarrier indexk once again, we useC(k)BC(gk, Pk) to denote the capacity region of

the BC on subcarrierk under a carrier specific sum power constraintPk andC(k)MAC(gk, P1,k, ..., PM,k)

to denote the capacity region of the dual MAC on subcarrierk under a set of carrier specific

power constraintsP1,k, ..., PM,k.

For the subsequent results we need the following definition.

Definition 2.2. The sum of sets is defined as

∑

n

Xndef=

z : z =

∑

n

xn, xn ∈ Xn ∀n

.

Lemma 2.3. [45, Thm. 2.1][46, Thm.1] With the stacked vector of channel gainsg = [gT1 , ..., g

TK]T

the capacity region of a set of parallel Gaussian broadcast channels under a sum power con-

straint P is given by

CBC(g, P) =⋃

∑Kk=1 Pk=P

K∑

k=1

C(k)BC(gk, Pk) (2.23)

whereC(k)BC(gk, Pk) was defined in(2.12).

No proof is given in [45] but the theorem is proven in by Li and Goldsmith in [46, Thm.1]

in the context of ergodic capacity of fading broadcast channels. Further, with the results of [36]

at hand it is very easy to proof Lemma2.3, since any set of parallel Gaussian BCs can be seen

as a diagonal MIMO BC.

Duality

Using the same argument as in Lemma2.3 it is easily shown that the BC capacity region of

parallel channels equals the union over MAC capacity regions.


Lemma 2.4. With the stacked vector of channel gainsg = [gT1 , ..., g

TK]T the capacity region of a

set of parallel Gaussian BCs under a sum power constraintP equals the union of MAC capacity

regions with individual power constraints summing up to thesame sum power

CBC(g, P) =⋃

∑Mm=1 Pm=P

CMAC(g, P1, ..., PM), (2.24)

where the MAC capacity region of parallel channels is given by

CMAC(g, P1, ..., PM) =⋃

∑Kk=1 Pm,k=Pm ∀ m

K∑

k=1

C(k)MAC(gk, P1,k, ..., PM,k)

andC(k)MAC(gk, P1, ..., PM) was defined in(2.15).

Proof.

CBC(g, P)(a)=

⋃

∑Kk=1 Pk=P

K∑

k=1

C(k)BC(gk, Pk)

(b)=

⋃

∑Kk=1 Pk=P

K∑

k=1

⋃

∑Mm=1 Pm,k=Pk


=⋃

∑Mm=1 Pm=P

⋃

∑Kk=1 Pm,k=Pm

K∑

k=1


The equalities (a) and (b) follow immediately from Lemma2.3and Theorem2.1, respectively.

The last step is just the commutation of summation and union. �

This equivalence opens the possibility to study problems for parallel BCs in the dual MAC

and then use the transformations in (2.21) to relate the solutions to the broadcast scenario.

2.3 System Model of Multiple-Antenna OFDM Broadcast and

Multiple-Access Channel

In order to cover MIMO-OFDM multi-user systems we extend thepreviously introduced system

model to the case where the base station is equipped withnB antennas and each user ownsnU

antennas. LetCm,l ∈ CnB×nU be thelth resolvable path of the array channel impulse response of

userm. Then

Hm,k =

L∑

l=1

Cm,l exp

(

− j2π(l − 1)(k− 1)K

)

m∈ M, k ∈ K (2.25)

yields the matrix valued channel transfer function, where[

Hm,k]

r,s is the channel between base

station antennar and antennas of the mth user on subcarrierk. The vector valued signal

2.3. System Model of Multiple-Antenna OFDM BC and MAC 15

+

+

+

+xk

x1,k

x2,k

xM,k

yk

y1,k

y2,k

yM,k

HH1,k

HH2,k

HHM,k

H1,k

H2,k

HM,k

nk

n1,k

n2,k

nM,k

Figure 2.3: Gaussian MIMO broadcast channel (left) and its dual MIMO multiple-access chan-nel (right) on subcarrierk

yDLm,k ∈ CnU received on subcarrierk by userm in the MIMO-OFDM downlink is then given by

yDLm,k = Hm,kxDL

k + nDLm,k ∀ m∈ M (2.26)

wherexDLk ∈ CnB is the signal transmitted on subcarrierk andnDL

m,k ∈ CnU , nDLm,k ∼ CN(0, I nU ) is

circular symmetric addive white Gaussian noise with normalized variance. The corresponding

uplink system equation with conjugate transposed channel matrices reads as

yULk =

M∑

m=1

HHm,kx

ULm,k + nUL

k (2.27)

whereyULk ∈ CnB is the signal received by the base station on subcarrierk, xUL

m,k ∈ CnU denotes

the signal transmitted by userm on subcarrierk andnULk ∈ CnB, nUL

k ∼ CN(0, I nB) denotes the

circular symmetric additive white Gaussian receiver noise. Note that we used the conjugate

transposed channel transfer function for reasons to becomeclear later. In analogy to (2.9) and

(2.10) the equations (2.26) and (2.27) are the system equations of a vector Gaussian BC and

MAC, respectively. Dropping the superscripts·DL and·UL both channels are depicted in Figure

2.3.

As in the scalar case we study the elementary MIMO BC and MAC first. Compared to

scalar BC the MIMO case is more complicated to handle. The main difference is that the input

and output of the channels are vector valued. The transmitter has more degrees of freedom to

choose a transmit strategy. A consequence is that the MIMO BCis non-degradedin general.

2.3.1 MIMO Gaussian Broadcast Channel

Consider the MIMO Gaussian BC on subcarrierk depicted in Figure2.3on the left. Dropping

the subcarrier indexk the received signal is given by

ym = Hmx + nm, ∀ m ∈ M (2.28)


whereHm ∈ CnU×nB is the matrix valued channel between the base station and user m, ym ∈ CnU

is the signal received by userm, x ∈ CnB is the transmit signal andnm ∼ CN(0, I nU ) is circular

symmetric additive white Gaussian noise. The BC in (2.28) is not a degraded broadcast channel

in general‡. For this reason superposition coding can not be used as a capacity achieving coding

technique. Instead, a coding technique widely calledDirty-Paper Coding (DPC)can be used.

The name stems from Costa’s famous paper entitledWriting on dirty paper[44], where he

considered a scalar Gaussian channel where the received signal is given by

y = x+ s+ z

with x being the power limited channel input,z being Gaussian noise ands being a Gaussian

distributed disturbance. The disturbances is assumed to benon-causallyknown to the trans-

mitter but not to the receiver. Costa showed that the capacity of the channel is the same as if the

disturbance was not present, or loosely speaking, knowing the dirt on a sheet of paper allows to

write on it. The result was generalized to vector-valued channels in [43]. Based on this result an

achievable rate region can be derived. In contrast to the single antenna case, the base station has

to encode the users’ signals in some order treating the transmit signals intended for previously

encoded users as non-causally known interference.

Letπ(·) be an arbitrary encoding order so that userπ(1) is encoded first followed by userπ(2)

and so on until userπ(M) is encoded last. Then the following ratesR1, ...,RM can be achieved

using DPC

Rπ(m)(H, π(·),Q1, ...,QM) = log

∣∣∣∣∣∣∣

I + Hπ(m)

∑

n≥m

Qπ(n)

HHπ(m)

∣∣∣∣∣∣∣

− log

∣∣∣∣∣∣∣

I + Hπ(m)

∑

n>m

Qπ(n)

HHπ(m)

∣∣∣∣∣∣∣

m ∈ M,

(2.29)

whereQm = E{

xmxHm

}

is the covariance matrix of the signal intended for userm, x =∑M

m=1 xm

is the signal transmitted in (2.28) andH = [HT1 , ...,H

TM]T denotes the matrix of stacked channel

matrices. Since the interference of already encoded users is rendered harmless using the DPC

strategy, only the covariance of the noise plus remaining interference occurs in (2.29).

It was shown in [36] recently that the above strategy and Gaussian codebooks indeed achieve

the capacity region.

Theorem 2.5. [36, Corollary 4] The capacity region of a MIMO Gaussian BC with channels

H = [HT1 , ...,H

TM]T and a sum power constraintP is given by

CBC(H, P) =⋃

Qm�0 ∀ m∈M: tr(∑Mm=1 Qm)≤P

RDPC(H,Q1, ...,QM) (2.30)

‡In fact there do exist degraded MIMO BCs. See [47] for a detailed study.


whereRDPC(H,Q1, ...,QM) is the rate region achievable with fixed transmit covariancematrices

RDPC(H,Q1, ...,QM)def= co

⋃

π∈Π

{

R : Rπ(m) = Rπ(m)(H, π(·),Q1, ...,QM) m ∈ M}

(2.31)

with Rπ(m)(H, π(·),Q1, ...,QM) given in(2.29) andΠ is the set of all possible M! permutations

onM.

The proof according to [36] is quite challenging using various intermediate steps. A simpler

converse proof was presented recently [48].

It is interesting to note that in contrast to the scalar BC theiterative process is shifted to

the transmitter. Moreover, all possible encoding orders might contribute to the capacity region

where as in the single antenna case a single optimum decodingorder can be identified.

2.3.2 MIMO Gaussian Multiple-Access Channel

Consider the MIMO MAC on subcarrierk depicted in Figure2.3 on the right. HereM users

transmit independent messages to a base station. The received signal is a linear superposition

of the M individual received signals corrupted by additive Gaussian noise at each antenna. The

system equation can be written as

y =M∑

m=1

HHmxm+ n, (2.32)

where the channel between usermand the receiver is given byHm ∈ CnU×nB, the vectorxm ∈ CnU

denotes the signal transmitted by this user andn ∼ CN(0, I nB) is circular symmetric additive

white Gaussian noise. In analogy to the single antenna case in Section2.2.2it is assumed that

each user’s power budget is limited toE{

xmxHm

}

≤ Pm.

In contrast to the broadcast channel the main structure of the MAC is preserved while gen-

eralizing from the scalar to the vector valued case. For any fixed set of transmit covariance

matricesQ1, ...,QM the achievable rate region is given by

RMAC (H,Q1, ...,QM) =

R :

∑

m∈IRm ≤ log

∣∣∣∣∣∣∣

I +∑

m∈IHH

mQmHm

∣∣∣∣∣∣∣

∀ I ⊆ M

. (2.33)

Like in the scalar case it can be easily verified that

f (I)def= log

∣∣∣∣∣∣∣

I +∑

m∈IHH

mQmHm

∣∣∣∣∣∣∣

, I ⊆ M (2.34)

is a submodular rank function. As a consequence (2.33) is a polymatroid as illustrated in Figure

2.2. However, the additional spatial degrees of freedom allow to achieve an uncountable number

of these polymatroids with the same set of individual power constraints. The MIMO MAC


capacity region is thus given by

CMAC(H, P1, ..., PM) =⋃

Qm�0:tr(Qm)≤Pm ∀∈M

RMAC(H,Q1, ...,QM) (2.35)

being a union of polymatroids. As for the scalar Gaussian MACeach vertex corresponds to a

specific successive interference cancellation order. Letπ(·) be a decoding order such that the

signal of userπ(M) is decoded first, followed by the signal of userπ(M − 1) and so on. Then

the achievable rates are given by

Rπ(m)(H, π(·),Q1, ...,QM) = log

∣∣∣∣∣∣∣

I +m∑

n=1

HHπ(n)Qπ(n)Hπ(n)

∣∣∣∣∣∣∣

− log

∣∣∣∣∣∣∣

I +m−1∑

n=1

HHπ(n)Qπ(n)Hπ(n)

∣∣∣∣∣∣∣

∀ m∈ M.

(2.36)

This is in analogy to the expression for the rates in the scalar MAC given in (2.17) . An alterna-

tive characterization of the MIMO MAC capacity region is thus given by

CMAC(H, P1, ..., PM) =⋃

Qm�0:tr(Qm)≤Pm ∀ m∈M

co

⋃

π∈Π

{

R : Rπ(m) = Rπ(m)(H, π(·),Q1, ...,QM)}

(2.37)

with Rπ(m)(H, π(·),Q1, ...,QM) defined in (2.36). It remains to point out the similarity of (2.37)

and the description of the MIMO BC capacity region given in (2.30).

2.3.3 Duality of MIMO Gaussian BC and MAC

Similar to the single antenna case a duality relationship exists for the MIMO BC and MAC.

Vishwanath et al. presented in [34] a duality result for the MIMO MAC capacity region and the

rate region of the MIMO BC achievable with DPC. Simple recursive transformations similar to

that in (2.20) and (2.21) were derived. It is interesting to note for historical reasons that at that

time the capacity region of the MIMO BC was not clear yet and the DPC region was just the

best known achievable rate region. The converse for the MIMOBC was proved later in [36].

Theorem 2.6. [34, Thm. 2.1] LetH = [HT1 , ...,H

TM]T be a set of channel realizations and let

CBC(H, P) be the MIMO BC capacity region under a sum power constraintP given in(2.30).

Let CMAC(H, P1, ..., PM) be the capacity region of the dual MIMO MAC given in(2.35) under

individual power constraintsP1, ..., PM. Then

CBC(H, P) =⋃

∑Mm=1 Pm≤P

CMAC(H, P1, ..., PM). (2.38)

Note that in contrast to the SISO case not only the channel gains but the channel realizations

are relevant, and the channel matrix of the dual MAC is hermitian transposed.


Let QBC1 , ...,QBC

M be a set of transmit covariance matrices for the BC, so that a superimposed

signal with covarianceQBC =∑M

m=1 QBCm is sent and a rate tupleR1, ...,RM is achieved using a

precoding orderπ(·). Further define the effective noise covariance matrices

ZBCπ(m) = I + Hπ(m)

m−1∑

n=1

QBCπ(n)

HHπ(m)

ZMACπ(m) = I +

M∑

n=m+1

HHπ(n)Q

MACπ(n) Hπ(n).

Let

He f fπ(m) = Uπ(m)Λπ(m)VH

π(m)

be the singular value decomposition of the effective channelHe f fπ(m) = ZBC−1/2

π(m) Hπ(m)ZMAC−1/2

π(m) . Then

the same set of rates can be achieved using the reverse decoding orderπ(·) and the transmit

covariance matrices

QMACπ(m) = ZBC−1/2

π(m) Uπ(m)VHπ(m)Z

MAC1/2

π(m) QBCπ(m)Z

MAC1/2

π(m) Vπ(m)UHπ(m)Z

BC−1/2

π(m) ∀ m∈ M. (2.39)

Conversely, for a given set of MAC covariance matrices the corresponding BC covariance ma-

trices achieving the same rate tuple are given by

QBCπ(m) = ZMAC−1/2

π(m) Vπ(m)UHπ(m)Z

BC1/2

π(m) QMACπ(m) ZBC1/2

π(m) Uπ(m)VHπ(m)Z

MAC−1/2

π(m) ∀ m∈ M. (2.40)

These transformations were also derived in [34]. Note that the transformations in (2.39) and

(2.40) simplify to those in (2.20) and (2.21) of the single antenna case withnU = nB = 1,

respectively. The difference is that the spatial degrees of freedom cause a base change w.r.t. to

the effective channel in each step.

With the results of this section at hand it becomes irrelevant whether a specific problem is

solved in the BC or the dual MAC since similar to the scalar case the solutions can be trans-

formed from the MAC to the BC and vice versa.

2.3.4 Parallel MIMO Gaussian BCs and MACs

Coming back to the initial system model consisting of a set ofparallel Gaussian MIMO BCs

the system model reads as

ym,k = Hm,kxk + nm,k, ∀ m ∈ M, k ∈ K (2.41)

whereHm,k ∈ CnU×nB is the matrix valued channel between the base station and user m on

subcarrierk, ym,k ∈ CnU is the signal received by userm on subcarrierk, xk ∈ CnB is the transmit

signal on subcarrierk and nm,k ∼ CN(0, I nU ) is circular symmetric additive white Gaussian


noise. The dual set of parallel Gaussian MIMO MACs is given by

yk =

M∑

m=1

HHm,kxm,k + nk ∀ k ∈ K , (2.42)

where the vectorxm,k ∈ CnU denotes the signal transmitted by this user andnk ∼ CN(0, I nB)

circular symmetric AWGN, each on subcarrierk. Stacking the receive signalsy = [yT1 , ..., y

TK]T ,

the transmit signalsxm = [xTm,1, ..., x

Tm,K]T and the noisen = [nT

1 , ..., nTK]T the set ofK parallel

vector MACs can be rewritten as

y =M∑

m=1

HHxm+ n

with block-diagonal channel matrices

Hm =

Hm,1 0 · · · 0

0 Hm,2 · · · 0. . .

0 · · · 0 Hm,K

∀ m ∈ M,

where we used0 to denote thenU ×nB all-zero matrix. The same obviously holds for the system

of K parallel vector BCs in (2.41). Any set of parallel MIMO BCs is a MIMO BC itself. As a

consequence duality holds also for parallel channels as in the single antenna scenario. This fact

will be exploited throughout the remainder of the thesis frequently. Problems will be solved in

the uplink, since the mathematical structure has favorableproperties such as the polymatroid

structure of elementary regions. The optimum transmit strategies can be simply transformed to

the BC using (2.40).

2.4 Summary

In this chapter we introduced the system model and presentedbasic results concerning the ca-

pacity regions of the scalar Gaussian BC and MAC as well as theMIMO Gaussian BC and

MAC. As shown, the results can be easily extended to the case of parallel channels. For scalar

as well as the vector valued multi-user channels duality relations exist which allow to trans-

form a transmit strategy achieving a certain rate tuple in the BC to a transmit strategy achieving

the same rate tuple in the dual MAC and vice versa while the sumpower is preserved. In

order to achieve arbitrary rate tuples in the MIMO BC Dirty-Paper Coding has to be applied

pre-compensating non-causally codewords of previously encoded users. This is due to the fact

that the MIMO Gaussian BC is not a degraded BC. For the scalar BC superimposing Gaussian

codewords combined with successive interference cancellation at the receivers achieves capac-

ity. DPC is thus a more general concept than Superposition Coding. In the MAC it is sufficient

to perform successive interference cancellation at each receiver in order to achieve capacity re-

2.4. Summary 21

gardless of the number of antennas. With these fundamental characterizations at hand we can

turn towards the problems of optimum resource allocation.

Chapter 3

Resource Allocation for Single Antenna

OFDM Systems with Time-Invariant

Channels

In general wireless channels are not static but fade over time. This is due to the mobility of

scatterers, transmitters and receivers. However, if the fading is sufficiently slow the channel can

be modeled as time-invariant over a certain time block and while it changes from one block to

another. This model leads to the commonly used so-calledblock-fadingassumption. Within

such an environment wireless communication systems have tosupport an increasing number of

services which are sensitive not only to the provided long-term data rate but also to delay. A

service might fail if a demanded rate can not be provided due to a deep fade during a certain

block. Hence, a power allocation being constant over various fading blocks is not suited to

deal with these conditions. The fading nature of the wireless channel thus calls for a transmit

strategyadapted to the current fading block channel conditions, given rate requirements and

resource constraints.

In this chapter we solve three fundamental resource allocation problems for the time-invariant

single antenna OFDM BC and MAC corresponding to the adaptiveoptimization within a sin-

gle fading block. Compared to MIMO-OFDM systems single antenna OFDM systems offer a

richer mathematical structure founded by three key properties:

1. First, single antenna parallel Gaussian BCs can be decomposed to a set of degraded broad-

cast channels as described in Section2.2.4.

2. Second, single antenna MACs are characterized by a generalized symmetric rank function

implying a useful contra-polymatroid structure (detailedin the appendix).

3. Third, a bijective mapping between power and rate exists on each of the parallel channels.

This allows for an optimization of the rate allocation instead of the power allocation.

23

24 Chapter 3. Resource Allocation for Single Antenna OFDM Systems

These facts play an important role in the following and can beexploited in order to derive

appealing algorithms.

The chapter is organized as follows. Section3.1introduces and motivates the three reference

problems. We then begin with a formulation relying on the broadcast scenario in Section3.2.

Section3.3contains an analysis in the dual MAC. Finally we discuss the question of an optimal

decoding order in Section3.4 and summarize the results in Section3.5. Parts of this chapter

have been also published in [1, 2, 3, 4].

3.1 Problem Statement

In the following the superscripts·BC and·MAC are dropped. For the sake of simplicity all prob-

lems are presented representatively for up- and downlink inthe broadcast scenario implying a

sum power constraint. For the multiple-access scenario thesum power constraint inC(g, P) has

to be substituted by individual power constraints yieldingC(g, P1, ..., PM).

Weighted Rate-Sum Maximization

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5

3

3.5

R1 [bps/Hz]

R2 [b

ps/H

z]

µ

Figure 3.1: Illustration of Problem1

The first problem is illustrated in Figure3.1exemplarily for the two user case and reads as

follows.

Problem 1 (Weighted Rate-Sum Maximization). For a given power budgetP, channel realiza-

tionsg and user specific weightsµ ∈ RM+ maximize a weighted sum of rates.

max µTR subj. to R ∈ C(g, P)

3.1. Problem Statement 25

Solving Problem1 yields a rate tuple on the boundary of the capacity regionC(g, P) with

supporting hyperplane normal vectorµ. SinceC(g, P) is a convex set the entire boundary of

the capacity region can be traced out by varying the weight vector µ. Note that throughput

maximization is just a special case withµ1 = ... = µM. Moreover, each rate tuple on the

boundary isPareto-optimal, i.e. given the power budgetP no individual rateRm can be increased

without decreasing another user’s rateRn n , m. Thus rate tuples lying on the boundary of

C(g, P) are calledefficient. They constitute the set of desired operating points and thus it is of

great interest to find the corresponding transmit strategies.

Another motivation for this problem stems from the study of system stability [49]. If we

assume ergodic data arrival processes for each user at the base station and an ergodic block

fading process with bounded channel states, thestability region, i.e. loosely speaking the set

of average arrival rate vectors for which a buffer overflow can be avoided, is maximized if

Problem1 is solved in each instantaneous system state [50, 51]. Being more precise, stability

is defined as the non-evanescence of queues [50, 51, 52]. In each step the users’ weightsµ

have to be chosen as their instantaneous queue lengths. In [52, 53, 54] these results were

extended to the MIMO case and in [5] a first study was carried out for OFDM downlink systems.

This perspective fits well to an operator’s point of view, forwhom system stability and overall

performance is the predominant objective. A disadvantage is that individual QoS-requirements

are not considered at all and the policy might yield stability-optimal but very unfair resource

allocations not able to support the demanded services. For this reason we change the perspective

and introduce Problem2.

Sum Power Minimization

In contrast to Problem1 we now consider QoS-necessities in terms of rate requirements. See

Figure3.2for an illustration.

Problem 2 (Sum Power Minimization). Given a vector of required ratesR ∈ RM+ and channel

realizationsg minimize the transmitted sum power such thatR is achieved:

min P subj. to R ∈ C(g,P)

The solution to Problem2 is the optimum power allocation and the sum powerP∗ corre-

sponding to the smallest capacity region such that the rate tuple R is contained inC(g,P∗). It

reflects a user-oriented or service-oriented perspective and thus can be seen as the anti-pole

to Problem1. Note that in the uplink, where each user has an individual power budget, the

objective can be generalized to a weighted sum of powers as will be detailed in Section3.3.1.

Both formulations, Problem1 as well as Problem2, fall short of the complex situation as

encountered in a more realistic setting. In order to to resolve this we turn towards Problem3

bringing together both perspectives.


0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5

3

3.5

R1 [bps/Hz]

R2 [b

ps/H

z]

R∗


Minimum Rates Aware Weighted Rate-Sum Maximization

Problem3 can be interpreted as a synthesis of the former two problem statements. It consists

of the same objective as Problem1 combined with the rate constraints reflecting QoS-demands

from Problem2.

Problem 3 (Minimum Rates Aware Weighted Rate-Sum Maximization). For a given power

budgetP, channel realizationsg, user specific weightsµ ∈ RM+ and minimum rate requirements

R ∈ RM+ maximize a weighted sum of rates:

max µTR subj. to R ∈ C(g, P)

R ≥ R

This formulation brings together the two previous contrasting perspectives. Hard rate con-

straints assure that the QoS-demands are met while the remaining freedom is used to take into

account the linear weighting factors of the objective. For the single user case a similar prob-

lem statement was chosen in the context of time-varying channels in [55]. There the authors

proposed the notion of service outage for a flat-fading channel. The ergodic sum rate was

maximized under a minimum rate constraint allowing a certain outage probability. Coming to

multi-user channels, in [56] Jindal and Goldsmith considered the problem of fading broadcast

channels with minimum rates under a long term power constraint. They showed that the system

can be interpreted as a broadcast channel with effective channels incorporating the rate con-

straints. The case of fading broadcast channels is an instance of parallel Gaussian broadcast

channels. The crucial difference is that in [55, 56] a rate constraint oneachof the channels par-

allel in time is imposed. This corresponds to a fixed rate on each subcarrier. In Problem3 the

rate constraint is over all parallel channels coupling the system which is more meaningful for

3.2. A Broadcast Channel Approach 27

systems with wideband character, where the channels are parallel in frequency. This problem

formulation is obviously more sophisticated than Problems1 and2 and the additional question

of feasibility arises.

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5

3

3.5

R1 [bps/Hz]

R2 [b

ps/H

z]

R2 ≥ 1bps/Hz

R1 ≥ 2bps/Hz

µ


3.2 A Broadcast Channel Approach

First we present an approach to solve the Problems1 and2 originally introduced in [45] by Tse

and further developed by Li and Goldsmith in [46]. We then generalize the used concepts to

solve Problem3 which was not studied in [45, 46]. The approach takes place in the broadcast

channel domain. Duality results which would have allowed totransform the problems to the

dual uplink were unknown at this time.

3.2.1 Characterization of the Capacity Region in the BC Domain

Tse’s characterization of the capacity region of parallel Gaussian BCs is elegant. The formula-

tion exploits the fact that each of the parallel channels is adegraded BC. Further he introduces

the notion ofmarginal utility functions, characterizing each user’s revenue via its derivative.

Recall the expression for the rate of userπ(·) in (2.12) on a single channel

Rπ(m) = log

(

1+gπ(m)pπ(m)

1+ gπ(m)∑

n<m pπ(n)

)

. (3.1)

It is achievable using superposition coding and a decoding order such that

π(·) : gπ(1) ≥ ... ≥ gπ(M)


and userπ(M) is decoded first, followed by userπ(M − 1) and so on. A key observation is that

the rate in (3.1) can be expressed as an integral over the derivative w.r.t. power as

Rπ(m) = log

(

1+gπ(m)pπ(m)

1+ gπ(m)∑

n<m pπ(n)

)

= log

1+ gπ(m)

∑

n≤m

pπ(n)

− log

1+ gπ(m)

∑

n<m

pπ(n)

=

∫ ∑

n≤m pπ(n)

∑

n<m pπ(n)

11

gπ(m)+ x

dx. (3.2)

As pointed out in [45] the expression in (3.2) has a rate splitting interpretation in the sense that

each user’s rate can be interpreted as a set of parallel data streams, each transmitted with power

dx. This observation leads to the following characterizationof efficient rate tuples.

Theorem 3.1 (Thm. 3.3 in [45]). Assume that the channel gains gm,k are distinct on each

channel, i.e. gm,k , gn,k for n , m and for all k. FurtherP is a fixed sum power budget. Then

the boundary of the capacity regionC(g, P) is given by

R∗(µ, P) :

M∑

m=1

µm = 1, µ ∈ RM+

with individual rates

R∗m(µ, P) =1K

K∑

k=1

∫

A(k)m

11

gm,k+ x

dx ∀ m∈ M (3.3)

and the integration is over the set

A(k)m =

{

z ∈ R+ : u(k)m (z) =

[

maxm

u(k)m (z)

]+}

. (3.4)

The functions u(k)m (z) are calledmarginal utility functionsand are given by

u(k)m (z) =

µm1

gm,k+ z− λ (3.5)

while the Lagrangian multiplierλ satisfies

K∑

k=1

[

maxm

(

µm

λ− 1

gm,k

)]+

= P. (3.6)

With this characterization ofC(g, P) at hand Problem1 can be solved. Alternatively, the

solution of Problem1 can be used to derive the characterization ofC(g, P) as introduced in

Theorem3.1.


3.2.2 Maximizing a Weighted Sum of Rates

First we consider Problem1. For the sake of brevity we present a slightly modified derivation of

the algorithmic solution proposed in [45] in the following. To this end we consider the weighted

rate-sum maximization problem on an arbitrary subcarrierk under a sum power constraintPk.

Using the rate-splitting characterization from (3.2) we have

fk(Pk) = maxp∈RM

+ :∑M

m=1 pm≤Pk

M∑

m=1

µmRm,k(p)

= maxp∈RM

+ :∑M

m=1 pm≤Pk

M∑

m=1

µπk(m)

∫ ∑

n≤m pπk(n)

∑

n<m pπk(n)

11

gπk(m),k+ x

dx, (3.7)

whereπk(·) is such thatgπk(1),k ≥ ... ≥ gπk(M),k. Assuming thatgm,k , gn,k for m, n for simplicity,

the solution of (3.7) is unique and the optimum power allocations are given by

pπk(m) =

min(Pk, z) −∑

n<m pπk(n) if m= 1, ...,M − 1

Pk −∑

n<m pπk(n) if m= M(3.8)

wherez is the solution toµπk(m)

1gπk(m),k

+ z=

µπk(m+1)

1gπk(m+1),k

+ z(3.9)

The interpretation behind this principle is that power is allocated to each user until a different

user has a higher contribution to the objective with the samemarginal power. This corresponds

to the intersections of the marginal utility functions in [45]. The case where channel gains

can be identical does not cause any problems, but leads to multiple optimal power allocations.

Using this characterization, Problem1 can be rewritten as

maxK∑

k=1

fk(Pk) subj. toK∑

k=1

Pk ≤ P. (3.10)

Lemma 3.2.For fixedµ ∈ RM+ and channel gainsgk ∈ RM

+ the function fk(Pk) in (3.7) is concave

in Pk ∈ [0,∞).

Proof. This is easy to see sincefk(Pk) can be expanded as

fk(Pk) =M∑

m=1

µπk(m)

∫ ∑

n≤m pπk(n)

∑

n<m pπk(n)

11

gπk(m),k+ x

dx

=

∫ Pk

0

max

m

µm1

gπk(m),k+ x

dx

=

∫ Pk

0

∂ fk(x)∂x

dx.


Algorithm 1 Downlink algorithm for Problem1(1) solve (3.6) for optimumλ∗ such thatP = P using bisectionfor k = 1 to K do

(2) order users according to their channel gainsgm,k

(3) calculate power allocations via intersections of (3.5)end for(3) determine optimum ratesR∗ using (3.3)

The first equality is just an alternative presentation incorporating the integration limits (and

thus power allocations) for each user. The derivative∂ fk(x)/∂x is the maximum of positive and

monotonously decreasing functions and thus positive and monotonously decreasing. It follows

that f is concave onR+. �

Consequently, the objective in (3.10) is convex whileP ∈ RK+ :

∑Kk=1 Pk ≤ P constitutes a

simplex. Hence the problem is a convex optimization problemand the Karush-Kuhn-Tucker

(KKT) conditions are necessary and sufficient for optimality. These are given by

max

m

µm1

gm,k+ Pk

− λ

+

= 0 ∀ k ∈ K (3.11)

K∑

k=1

Pk ≤ P (3.12)

λ ≥ 0 (3.13)

λ

K∑

k=1

Pk − P

= 0. (3.14)

In order to get the individual power budgets, (3.8) can be used.

The characterization in Theorem3.1 reflects exactly the KKT-conditions for any set of

weightsµ ∈ RM+ summing up to one achieving the boundary ofC(g, P). Solving (3.11) for Pk

and substituting it into (3.12) yields the power constraint (3.6). The marginal utility functions

given in (3.5) characterize the revenue of each user’s rate to the Lagrangian function. Note that

Tse does not make use of the convexity properties of the problem in [45]. However, as shown

above the characterization can be also easily derived usingthe KKT-conditions.

The following algorithmic solution for Problem1 was proposed in [45]. Solve (3.6) for the

optimal Lagrangian multiplierλ∗. This can be done by bisection since the left hand side of (3.6)

is monotonically decreasing inλ. Onceλ∗ is found, solve for (3.4). To this end the intersections

of the marginal utility functions have to be found on each subcarrier. As shown in [45], they are

ordered in decreasing order of the channel gainsgm,k, so thatA(k)πk(M) neighboursA(k)

πk(M−1) and so

on. Further, ifA(k)πk(m) = ∅ thenA(k)

πk(m−1) = ∅. The procedure is summarized in Algorithm1.


C(g, P)

G(g)

P

P

R1 R2

(0, 0)

Figure 3.4: Illustration of the setG(g) and exemplaryC(g, P) for M = 2 users

3.2.3 Sum Power Minimization

The characterization in Theorem3.1 is also suited to solve Problem2. This is not obvious at

first glance.

The enhanced setG(g)

First we introduce the set

G(g)def=

{

(R,P) ∈ RM+1+ : R ∈ C(g,P)

}

. (3.15)

It contains all admissible rate-power tuples. An importantstatement about the shape ofG(g)

provides the subsequent lemma.

Lemma 3.3([45]). The setG(g) is a convex set.

This fact is illustrated exemplarily in Figure3.4 for the case of two users. Each horizontal

slice corresponds to a capacity regionC(g, P) for a fixed sum powerP. The convexity ofG(g)

provides much more structure than the convexity of each capacity region.

As a consequence it is sufficient to find Lagrangian multipliersµ ∈ RM+ such that the rate

requirementsR are met with equality andP∗ is a solution to

minP∈[0,∞)

λP− µTR(P), (3.16)

whereλ can be chosen arbitrarily. Geometrically speaking, the solution is a point on the surface

of G(g) with tangent hyperplane normal vector [µT − λ]T , where the elementsµ1, ..., µM of the

normal vector are chosen such that the rate constraints are met.


Algorithm 2 Downlink algorithm for Problem2

(1) initialize µ(0) = 0while desired accuracy not reacheddo

for m= 1 to M do(2) adjustµ(n+1)

m so thatRm = Rm using bisectionend for

end while

The algorithmic solution suggested in [45] consists of the following procedure: Initialize

the Lagrangian factors corresponding to the rate constraints asµ(0) = 0. Begin with user 1

and increaseµ(1)1 such thatR1 = R1. To this end the expressions in (3.3) have to be evaluated.

Repeat this procedure cyclically over the set of usersM. In the limit the Lagrangian multipliers

limn→∞ µ(n) yielding a solution to Problem2 are attained [45, Prop. 3.5]. The convergence proof

is based on a monotonicity property, stating that if the factor µm is increased the rateRm will

increase or remain the same for userm while for all other usersn , m the rateRn will decrease

or remain unchanged. The procedure is summarized in Algorithm2.

Although Algorithm2 looks very simple a disadvantage is that for any choice of theLa-

grangian multipliersµ in step (2) the rate expressions have to be evaluated in orderto check the

constraints. To this end all relevant intersections of the marginal utility functions determining

the power allocation have to be examined. Since in each iteration n the optimumµ(n)m has to be

found via bisection this involves a considerable computational complexity. A second disadvan-

tage is that for any finite iteration indexn the solutionµ(n) yields an infeasible rate allocation.

Feasibility is attained asn→ ∞.

3.2.4 Introducing Minimum Rates

The combination of both previous formulations, i.e. the limited sum power budget and mini-

mum rate requirements, leads to Problem3. This case was not considered in [45]. Nevertheless,

the provided framework together with some additional results allows for a solution of this prob-

lem as well. First we say some words on convexity and feasibility of the problem.

Convexity and Feasibility

In contrast to Problems1 and2 the question of feasibility arises, since the feasible set might be

empty. Recalling the problem formulation and the illustration in Figure3.3, the feasible set is

given by

R f (g, P) ={

R ∈ RM+ : R ∈ C(g, P),R ≥ R

}

(3.17)

Lemma 3.4. Problem3 is a convex problem. Given minimum rate constraintsR, a sum power

budgetP and channel gainsg it is feasible if and only if

P∗P2≤ P, (3.18)


Algorithm 3 Downlink minimum rates algorithm for Problem3

(1) initialize µ(0) = µ

while desired accuracy not reacheddofor m= 1 to M do

(2) solveR(n) = arg max µ(n)TR subj. to R ∈ C(g, P)

using Algorithm1if R(n)

m < Rm then(3) increaseµ(n)

m such thatR(n)m = Rm using bisection

end ifend for

end while

where P∗P2is the solution to Problem2 with the same rate requirements and channel gains.

Proof. With Rmf =

{

R ∈ RM+ : Rm ≥ Rm

}

the feasible set in (3.17) can be rewritten as the inter-

section of the capacity region withM half spaces

R f (g, P) = C(g, P)⋂

(∩Mm=1Rm

f ).

SinceR f (g, P) is the intersection of convex sets it is a convex set as well.The objective∑M

m=1 µmRm is linear and thus Problem3 is a convex optimization problem.

The second part is easy to see. The non-emptiness of the feasible set is necessary and

sufficient for feasibility. The feasible set is nonempty if and only if R ∈ C(g, P). By definition

of the capacity region this is equivalent to the statement that the rate tupleR is achievable with

powerP∗P2≤ P. �

Algorithmic Solution

We focus on the development of an algorithm now. In fact, similar to the power minimization

algorithm Problem3 can be solved by adjusting the Lagrangian multipliers in a sequential

manner: If the rate constraint of userm is violated his Lagrangian multiplier is increased such

that the target rate is met. This process is repeated cyclically over the set of users until the

process converges. The procedure is summarized in Algorithm 3.

In order to prove that Algorithm3 indeed converges to the global optimum we need the

following Lemma. It characterizes the behavior of the ratesgiven in (3.3) as a function ofµ.

It is worth noting that the situation is different from that in [45, Lemma 3.6] sinceλ has to be

adjusted due to the fixed sum power budget.

Lemma 3.5. For all m and a fixed sum powerP, if the mth componentµm of the Lagrangian

vectorµ is increased and the other components are held fixed, the rateRm(µ) given in (3.3)

remains the same or increases while Rn(µ) remains the same or decreases for all n, m.


Proof. The rates in (3.3) depend only on the size of the corresponding sets

A(k)n ≡

{

z ∈ RM+ : u(k)

n (z) =[

maxi∈M

u(k)i (z)

]+}

.

An increase ofµm causes the setsA(k)n to shrink or does not affect them for alln , mand for all

k. This is easy to see sinceµm occurs in the numerator ofu(k)m (z).

It remains to show that an increase ofµm can not reduce the set size of the same userm on

any subcarrier. Distinguish two cases:

∄k : m= arg maxn∈M

z0(n) : u(k)n (z0) = 0,

i.e. the marginal utility function of userm doesnot constitute the last intersection on any

subcarrierk, the parameterλ remains unchanged and hence the setsA(k)m can not shrink. If the

marginal utility function of userm constitutes the last intersection on at least one subcarrier k,

i.e.

∃k : m= arg maxn∈M

z0(n) : u(k)n (z0) = 0,

thenλ increases to meet the sum power constraint. The intersection of the marginal utility

functionu(k)m of userm is shifted by any∆µm and∆λ about the same offset

∆z=λ∆µm− µm∆λ

(∆λ + λ)λ

independent of the carrierk. This shift is equal for all carriers and nonnegative, sincethe power

constraint has to be met and all other carriers

∀ k : m, arg maxn∈M

z0(n) : u(k)n (z0) = 0

experience a power decrease. Hence, the rate of usermcan not decrease. �

With Lemma3.5at hand we can prove the following result assuring convergence.

Theorem 3.6. If Problem3 is feasible, then Algorithm3 converges to a stationary pointR∗,

which is the global optimum of Problem3.

Proof. Assume that Problem3 is feasible. Due to the convexity of the problem it suffices to find

a set of Lagrangian multipliersµ′ ∈ RM+ andλ ∈ R+ such that the rate constraintsR are fulfilled

with equality or are not active and the sum power budgetP is met. Consider the Lagrangian of

Problem3:

L(R, µ′, λ) = µTR − λ(P− P) − µ′T(R − R)

= (µ + µ′)TR − λ(P− P) − µ′TR (3.19)


The evaluation of the dual

g(µ′) = maxR∈C(g,P)

(µ + µ′)TR − µ′TR (3.20)

can be done by Algorithm1, since (3.20) is an affine version of Problem1. Givenµ = µ + µ′,

the set of equations characterizing the optimum rate and power allocation is given in (3.3)-(3.6).

The remaining question is how to varyµ′ ∈ RM+ (and thusµ) in order to obtain the optimal value

µ∗ of µ.

From Lemma3.5 it is known thatRm(µ) is a monotone function inµm for any fixed sum

powerP. Further, since the problem is assumed to be feasible we have

limµm→∞

Rm(µ) = Rsum ≥ Rm ∀ m ∈ M

whereRsum is the single user water-filling solution for userm

Rsum = max

p:∑

kpk≤P

1K

K∑

k=1

log(1+ pkgm,k).

Hence in iterationn+ 1 one can findµ(n+1) such thatR(n+1)m = Rm in case thatR(n)

m < Rm. Using

Lemma3.5, µ(n) is a component-wise monotone sequence. Define a mappingU representing

the update of the sequenceµ(n). Lemma3.5 guarantees thatU is order preserving. Starting

with µ(0) = µ it holdsUn(µ(0)) ≤ µ∗ for anyn, whereµ∗ ≥ µ is the solution such that the KKT

conditions are fulfilled anda ≥ b refers to component-wise greater or equal. Hence,{µ(n)} is a

component-wise monotone sequence bounded from above converging to the limiting fixed point

µ∗ such that the KKT conditions are fulfilled and the associatedrate vectorR∗(µ∗) is achieved.

This concludes the proof. �

To summarize, Algorithm3 consists of a sequence of weighted sum rate-maximizations

where the weightsµ change in an iterative manner: The violation of a rate constraint in step

(2) leads to an increase of the corresponding weight factor in step (3) which in turn leads to a

violation of other rate constraints. Figure3.5 illustrates the convergence process forK = 256

exemplary parallel random channels andM = 4 users. The monotonicity of the the sequenceµ(n)

is obvious. It remains to point out that each iterationn in Figure3.5 consists of four bisection

processes, each adjusting a weightµ(n)m . Note that similar to Algorithm2, Algorithm 3 has the

aggravating disadvantage that it produces a sequence ofinfeasibleresource allocations violating

at least one rate constraint at a time. Feasibility is attained asn→ ∞.


5 10 15 20 25 30 35 400.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Iterations

R [b

ps/H

z]

user 1user 2user 3user4

5 10 15 20 25 30 35 400.2

0.25

0.3

0.35

0.4

0.45

0.5

Iterations

µ

user 1user 2user 3user 4

Figure 3.5: Convergence behavior of Algorithm3: R (top) andµ = µ + µ′ (bottom) for asystem withM = 4 users,K = 256 at 7.5dB with µ = [0.25 0.29 0.3 0.2]T and required ratesR = [0 1.5 0.75 0.8]Tbps/Hz

3.3 A Multiple-Access Channel Approach

In this section we present an approach towards the three reference problems formulating them

in the dual multiple-access channel. In principle the rate splitting representation used in the

broadcast channel approach holds for the dual MAC and an analog framework can be derived

[57]. In [57, 58], where the rate splitting interpretation was developed, fading MACs were

studied in detail. Since fading MACs fall into the class of parallel channels, the results apply to

the case studied in this thesis. However, we will use a completely different approach and extend

the results. The used approach allows to derive a class of algorithms which can be interpreted

as iterativerate water-filling algorithms. Apart from the nice interpretation they have certain

algorithmic advantages.

3.3. A Multiple-Access Channel Approach 37

0 200 400 600 800 1000 12000

200

400

600

800

1000

1200

1400

1600

1800

P1 [linear]

P2 [l

inea

r]

P(g, R)λ

Figure 3.6: Exemplary power regionP(g, R)

We begin with Problem2, the sum power minimization problem, for reasons to become

clear later.

3.3.1 Weighted Sum Power Minimization

In contrast to the downlink scenario each user has a separatepower budget in the MAC. Hence

we consider the following modified version of Problem2:

minλTp subj. to R ∈ C(g, p1, ..., pM). (3.21)

whereλ ∈ RM+ is an arbitrary vector of power weights andp = [p1, ..., pM]T is the vector of the

users’ transmit power budgets and all variables refer to themultiple-access channel.

In order to give (3.21) a physical meaning, we introduce the set of powers

P(g, R) ={

p ∈ RM+ : R ∈ C(g, p1, ..., pM)

}

. (3.22)

It contains all individual power budgets which allow to support the given demanded vector of

ratesR. Under the assumption thatgm,k > 0 for eachm and at least one subcarrierk the set

(3.22) is convex and non-empty. The optimization problem (3.21) yields a power tuple on the

boundary of (3.22) characterized by a unique tangent hyperplane with normal vectorλ. By

varying the components ofλ the entire boundary of (3.22) can be traced out. Further all power

tuples on the boundary are Pareto-optimal power allocations. For an illustration see Figure3.6.

At first glance it is difficult to say something about the optimum transmit strategy. Astraight


forward reformulation of (3.21) yields

minπ,p

M∑

m=1

λm

K∑

k=1

pm,k

subj. to1K

K∑

k=1

log

(

1+pπ(m),kgπ(m),k

1+∑

n<m pπ(n),kgπ(n),k

)

≥ Rπ(m) ∀ m

(3.23)

where the optimization takes place over the power allocation and theM! possible decoding

orders. The expressions for the individual rates and thus the constraints are not convex in

p. Moreover, the polymatroid structure of the rate region achievable with power allocationp

assures that the rate tupleR is achievable with minimum powerusing some optimal decoding

order π(·) on all subcarriers. However, this optimal decoding order is unknown so far. Both

facts complicate the problem significantly. An alternativeis to take into account the 2M − 1

constraints characterizing the MAC capacity region directly: This yields a convex problem with

a number of constraints exponential inM [59]. We study this approach in detail for the MIMO-

OFDM case in Section5.2.

In the following we recast the problem using the fact that capacity can be achieved using

individual decoding orders on each parallel channel. According to (2.17) the rate of userπk(m)

on subcarrierk using decoding orderπk(·) is given by

Rπk(m),k = log

(

1+gπk(m),kpπk(m)

1+∑m−1

n=1 gπk(n),kpπk(n)

)

. (3.24)

Solving recursively forpπk(m),k yields after some manipulations

pπk(m),k =1

gπk(m),k

(

exp(

Rπk(m),k) − 1

)

exp

∑

n<m

Rπk(n),k

. (3.25)

Given a set of decoding ordersπk(·), for k = 1, ...,K and the expression in (3.25) the objective

in (3.21) can be formulated in terms of the individual user’s rates:

g(R) =K∑

k=1

M∑

m=1

λπk(m)1

gπk(m),k

(

exp(

Rπk(m),k) − 1

)

exp

∑

n<m

Rπk(n),k

. (3.26)

In general (3.26) is not a convex function onR ∈ RMK+ . Now choose on each subcarrierk a

decoding order according to

πk(·) :λπk(1)

gπk(1),k≥ ... ≥

λπk(M)

gπk(M),k(3.27)

where userπk(M) is decoded first followed by userπk(M −1) and so on. Then the objective can


be rewritten as

g(R) =K∑

k=1

M∑

m=1

λπk(m)

gπk(m),k

(

exp(

Rπk(m),k) − 1

)

exp

∑

n<m

Rπk(n),k

=

K∑

k=1

M∑

m=1

cm,k exp

∑

n≤m

Rπk(n),k

+ c0,k (3.28)

with

cm,k =

− λπk(1)

gπk(1),km= 0

λπk(m)

gπk(m),k− λπk(m+1)

gπk(m+1),km= 1, ...,M − 1

λπk(M)

gπk(M),km= M

For this specific choice of the decoding orders the coefficients are nonnegative form= 1, ...,M.

Hence the objective given in (3.28) consists of a sum of log-convex addends and thus is log-

convex. This yields a convex optimization problem. Furtherthe constraints areseparableand

block coordinate methods can be applied [60]. The following lemma states that the chosen

decoding orders are indeed optimal. The result was originally derived in [58, p. 2819]. For the

sake of completeness we proof it in the following.

Lemma 3.7. [58, p. 2819] Given(3.21) with any set of weightsλ ∈ RM+ , the decoding orders

in (3.27) are the optimum decoding orders in order to achieve the minimum weighted sum of

powers.

Proof. The proof involves some definitions and results concerning polymatroids which can

be found in the Appendix. Consider the rate tuple achievableon subcarrierk given a power

allocationp1,k, ..., pM,k. Using (2.15) it is characterized by a polymatroid

C(k) (gk, p1,K, .., pM,k)

=

R :

∑

m∈IRm,k ≤ f (I) ∀ I ⊆ M

with submodular rank function

f (I) = log

1+∑

m∈Igm,kpm,k

, I ⊆ M. (3.29)

According to DefinitionA.6 in the Appendix the rank function in (3.29) is generalized symmet-

ric. As a consequence using LemmaA.7 the set of transmit powers weighted with the channel

gainsDk(R1,k, ...,RM,k) which achieve a certain set of ratesR1,k, ...,RM,k is a contra-polymatroid.

With ym,k = gm,kpm,k andyk = [y1,k, ..., yM,k]T we have

Dk(R1,k, ...,RM,k) =

yk ∈ RM

+ :∑

n∈Syn,k ≥ exp

∑

n∈SRn,k

− 1 for all S ⊆ M.


Thus the objective in (3.21) can be written as

minDk(R1,k,...,RM,k) ∀ k

K∑

k=1

M∑

m=1

λm

gm,kym,k subj. to

1K

K∑

k=1

Rm,k ≥ Rm m∈ M

which is equivalent to

K∑

k=1

minDk(R1,k,...,RM,k)

M∑

m=1

λm

gm,kym,k subj. to

1K

K∑

k=1

Rm,k ≥ Rm m ∈ M.

Now TheoremA.9 from the appendix applies to each addend. That is, on each subcarrierk the

optimum permutation is given by (3.27), and thus the claim follows. �

Remark 3.8. It is interesting to note that any set of decoding orders not fulfilling (3.27) does

not result in a convex problem. In particular, choosing a common decoding order on all parallel

channels does not yield a convex problem.

Remark3.8is quite counter-intuitive. It turns out that there exists indeed a globally optimal

decoding order which can be applied to all subcarriers. But even for this optimal decoding

order the problem is not convex. Although seeming contradictory, both facts fit together and

can be explained by the structure of the optimum power allocation. Users get zero power on

these subcarriers which cause the objective to be non-convex for the globally optimum ordering.

Section3.4 is devoted to a detailed discussion of this fact.

The Lagrangian is given by

L(R, µ,ψ) = g(R) +M∑

m=1

µm

Rm−1K

K∑

k=1

Rm,k

−M∑

m=1

K∑

k=1

ψm,kRm,k

whereµ ∈ RM+ andψ ∈ RMK

+ are the vectors of Lagrangian multipliers corresponding tothe in-

dividual rate constraints and non-negativity constraints, respectively. The Karush-Kuhn-Tucker

(KKT) conditions are necessary and sufficient for the optimality of a given set of rates and read

as

M∑

s=m

cs,k exp

∑

n≤s

Rπk(n),k

= µπk(m) + ψπk(m),k (3.30)

1K

K∑

k=1

Rm,k − Rm ≥ 0

µm

1K

K∑

k=1

Rm,k − Rm

= 0 µm ∈ R+

ψm,kRm,k = 0 ψm,k ∈ R+, Rm,k ≥ 0


A simple reformulation of (3.30) yields

Rm,k =[

log(µm) − nm,k]+ ∀ m, k (3.31)

with

nπk(m),k = log

M∑

s=m

cs,k exp

∑

n<sn,m

Rπk(n),k

(3.32)

and whereµm is chosen such that1K

K∑

k=1

Rm,k = Rm.

Interestingly (3.31) can be interpreted asrate water-fillingwith effective noise levels given in

(3.32). We can perform a cyclic iterative adaption of the user’s rates where each elementary step

consists of a water-filling procedure. The corresponding steps are summarized in Algorithm4.

Theorem 3.9. Algorithm4 converges to a stationary pointR∗ which is the global optimum of

(3.21).

Proof. The problem is convex and the constraints are separable. Theobjective is twice continu-

ously differentiable and in each step the water-filling solution is unique. Thus block coordinate

descent methods converge to a stationary point [60, Prop. 2.7.1]. Since Algorithm4 is a special

case of a block coordinate descent method and further due to convexity each stationary point is

a global optimum, the claim follows. �

Algorithm 4 Iterative rate water-fillingwhile desired accuracy not reacheddo

for m= 1 to M do(1) compute the coefficientsnm,k (3.32) for userm and allk(2) do rate water-filling according to (3.31) so thatRm is met

end forend while(3) calculate corresponding power allocation from rate allocation using (3.25)

Note that the algorithm was independently derived in [1] and [61] very recently. The conver-

gence is illustrated in Figure3.7. In order to compare the convergence behavior the algorithm

proposed in [62] is also included in the figure: The authors proposed to use a fixed decoding

order on all subcarriers. As described above this results ina non-convex problem, even for the

optimal decoding order. There is no convergence proof for this algorithm up to now. It can

be observed that the proposed algorithm converges very quickly even for a system with 256

parallel channels. The reason is the log-convexity of the objective function in the presented

formulation. In contrast, the convergence behavior for a fixed decoding order is not predictable.


10 20 30 40 50 60 70 80 90 1000.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7x 10

5

Iterations

P [l

inea

r]

3→2→12→1→33→1→2individual

Figure 3.7: Comparison of convergence behavior for exemplary random channel withM = 3users,K = 256,λ = [1 1 1]T andR = [1.5 1.5 1.5]T bps/Hz. Fixed decoding orders refer tothe algorithm presented in [62] and individual decoding orders (red stars) indicate the presentedrate water-filling algorithm (Algorithm4)

The chosen objective withλ = [1 1 1]T corresponds to sum power minimization, a case

especially of interest for broadcast scenarios. Due to duality the resulting power allocation can

be transformed to the BC yielding the same minimum sum power.

3.3.2 Extension to Minimum Rate Requirements

The solution presented for weighted sum power minimizationcan also be generalized in or-

der to face the two remaining problems of weighted rate sum maximization with and without

minimum rate requirements.

To this end consider the convex setG(g) defined in (3.15) for the broadcast scenario. We

can introduce an equivalent set for the MAC case using the vector of power budgetsP ∈ RM+ :

Gind(g) = {(R,P) : R ∈ C(g,P1, ...,PM)}

As a direct consequence of the concavity of the log-functionthe setGind(g) is a convex set [45].

The subscript·ind stands for individual power constraints. In contrast toG(g) the set is hard to

visualize since even for the case of two users we have (R,P) ∈ R4+.

Consider the problem

max µTR − λTP subj. to (R,P) ∈ Gind(g). (3.33)

Then for any point on the boundary ofGind(g) there exist multipliersµ ∈ RM+ andλ ∈ RM

+

such that this point is a solution to (3.33) [45, Lemma 3.10]. Thus it remains to find the opti-


Algorithm 5 Water-filling based minimum rates algorithm

(1) choose initial Lagrangian factorsλ(0)

while power constraintsP are not metdowhile desired accuracy not reacheddo

for m= 1 to M do(2) compute the coefficientsnm,k (3.32) for userm with decoding order specified in(3.27)(3) do water-filling with fixed level log(¯µm/λ

(n)m ) according to (3.31)

if Rm < Rm then(4) choose water-filling level log

(

(µm + µ′m)/λ(n)

m

)

with µ′m > 0 such thatRm = Rm

end ifend for

end while(5) updateλ(n+1) according to standard ellipsoid rule or any other sub-gradient method fornon-differentiable convex optimization

end while

mum multipliers such that the KKT-conditions of the particular problem are fulfilled. Also the

weighted sum power minimization problem studied in the previous section can be interpreted in

this framework: The power weightsλ are given in advance, where as the Lagrangian multipliers

µ are chosen in terms of the current water-filling level.

The Lagrangian of the studied weighted rate sum maximization problem with minimum

rates is given by

L(R,P, µ′, λ) = µTR + µ′T(R − R) − λT(P− P).

As in the BC case settingµ = µ+µ′ reveals the intimate connection to the general formulationin

(3.33). As a consequence we can use the following procedure summarized in Algorithm5 to ob-

tain the optimum. Each step consists of an iterative water-filling procedure for fixed Lagrangian

multipliersλ(n) corresponding to the power constraints. The rate constraints are kept implicit

which is equivalent to evaluate the dual over the constrained setGind(g) ∩{

(R,P) : R ≥ R}

at

λ(n).

g(λ(n)) = maxGind(g)∩{(R,P):R≥R}

µTR − λT(P− P).

Since the dualg(λ) is convex inλ it remains to find the optimum Lagrangian multipliers by any

convex optimization method suited for the optimization of non-differentiable objectives. For

example, the standard ellipsoid method can be used [63, 64]. The same algorithm can be used

if the power budgets are limited by a sum power constraint as it is the case in the downlink. In

this case step (5) reduces to a simple bisection process.

We have the following theorem that ensures convergence.

Theorem 3.10. If Problem3 is feasible then Algorithm5 converges to a stationary pointR∗,

which is the global optimum of Problem3.


Proof. First we show that in case of feasibility for any fixedλ the inner loop converges: for any

fixedλ, the set of rates

R = {Rm,k : Rm,k ≤ maxm

Rm m∈ M}. (3.34)

is a compact convex set. The objective functiong(R) = µTR is continuously twice differentiable

onR and the rate constraints are separable. In each partial step, the unique solution to the convex

optimization problem of one specific userm (i.e. a coupled subset of variables) is found. Thus

using [60, Prop. 2.7.1], convergence is assured.

The outer loop converges per definition since the dual is convex and we can apply any

standard method. Consequently, Algorithm5 converges to a stationary pointR∗ being the global

optimum. �

Corollary 3.11. Algorithm5 can be used to solve the problem

max µTR subj. to R ∈ C(g, P). (3.35)

This follows immediately from the fact that (3.35) is a special case of Problem3 setting

R = 0M.

3.4 Individual vs. Global Successive Decoding Order

An important and somewhat surprising property of parallel Gaussian MACs and BCs is the

optimal decoding order. The same observation being consistent with the results presented here

was made by Jindal et al. in the context of ergodic MAC and BC capacity regions in [38]. We

focus on the MAC case first, and generalize the observation toparallel BCs then.

3.4.1 Optimum Decoding Order for Parallel MACs

Each point on the boundary of the capacity region of parallelGaussian MACs can be achieved

using successive interference cancellation with individual stripping order on each channel de-

fined in (3.27). Interestingly, there always exists a stripping orderπ(·) determined by the initial

weights and Lagrangian multipliers which can be applied to all parallel channels yielding the

same rate tuple. This is of considerable interest since coding is often performed over the par-

allel systems (e.g. subcarriers in OFDM) in practice so thatthe application of an individual

successive decoding order is not practical or even impossible.

With the vector

µ = µ + µ′ (3.36)

the solution to Problem3 is also a solution to (3.33), whereλ is chosen such that the power

constraints are met. Due to the polymatroid structure and TheoremA.8 from the appendix the

3.4. Individual vs. Global Successive Decoding Order 45

decoding order

π(·) : µπ(1) ≥ ... ≥ µπ(M), (3.37)

where userπ(M) is decoded first followed buy userπ(M−1) and so on, achieves the correspond-

ing vertex and thus is optimal. Referring to [57, Lemma 3.10], (R,P) is a solution to (3.33) if

and only if it is a solution to theK individual problems

max µTRk − λTPk subj. to Rk ∈ C(gk,Pk) (3.38)

for all k, whereRk = [R1,k, ...,RM,k]T andPk = [P1,k, ...,PM,k]T are the vectors of rates and powers

on thekth channel respectively,∑K

k=1 Pm,k = Pm and∑K

k=1 Rm,k = Rm for all m. ThusRk must be

a vertex of the polymatroid constituting the elementary region on subcarrierk achievable with

decoding order (3.37). At first glance it seems somehow strange that the global decoding order

given by (3.37) as well asthe set of decoding orders defined in (3.27) are optimal. However,

this is the case. The reason is the generalized symmetry of the rank function yielding the contra-

polymatroid structure. A consequence is that

pm,k > 0 only if µm ≥ µn ∀ n :λn

gn,k≥ λm

gm,k.

In other words, on subcarrierk power is allocated only to usersn whose factorsµn fit to the

corresponding optimum decoding order deterimined by the channel gainsgk and weightsλ.

3.4.2 Optimum Decoding Order for Parallel BCs

For the case of parallel BCs we have a similar result. The difference to the MAC is that the

power is constrained by a sum power budget instead of individual budgets and we haveλm = 1

for all m. Due to the non-degradedness of the system capacity can be achieved using DPC and

a common precoding order specified by (3.37). However, capacity can be also achieved using

Superposition Coding and acommondecoding order given by (3.37) where userπ(1) is decoded

first followed by userπ(2) and so on. This is interesting since Superposition Coding implies

a completely different coding strategy than DPC. Similar to (3.38) consider the elementary

problem

max µTRk − λPk subj. to Rk ∈ C(gk,Pk). (3.39)

The degradedness of sub-channelk yields

pm,k > 0 only if µm ≥ µn ∀ n : gn,k ≥ gm,k.

In other words, any user gets non-zero power on subcarrierk only if all users with stronger

channel gains on this subcarrier have smaller weight factors. This allows to apply the same suc-


0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5

3

3.5

R1 [bps/Hz]

R2 [b

ps/H

z]

R1 > 2 bps/Hz

µ

R2 > 1 bps/Hz

µ = µ + µ′

Figure 3.8: Illustration of the optimum decoding order given byµ = µ + µ′

cessive decoding order on all subcarriers. An intuitive explanation is the following: Considering

an arbitrary subcarrierk and any pair of users with channel gaingm,k > gn,k and a fixed amount

of power, the sum capacity point for this subregion lies on the axis of userm, i.e. the user with

stronger channel. This means that the boundary of the entiresub-region can be traced out with

one specific ordering of weights, while for the inverse ordering of the weights the solution of a

weighted rate-sum maximization will lie on an axis yieldingzero power for one of the users.

Note that the optimum global decoding order (3.37) involves the initial weightsµ and the

Lagrangian multipliersµ. This means that the optimum decoding order is not determined by

the initial weightsµ alone. This fact is illustrated in Fig.3.8. The tangent point forµ is not

feasible. The Lagrangian multipliers modify the normal vector (and thus the decoding order)

such that the rate constraints are fulfilled. If there are no rate constraints, the initial weights

determine the optimum decoding order sinceµ = µ.

Another interesting observation is that the optimum globaldecoding order changes over

SNR, if minimum rates are required. This is due to the fact that the rate prices vary according

to the sum power budget: They decrease monotonously with an increase of the sum powerP.

This behavior can be observed in Fig.3.9. Below a transmit SNR of 6.8 dB the problem is

infeasible, i.e. the minimum rate requirements can not be fulfilled. With increasing SNR the

optimum global decoding order changes twice. As the Lagrangian multipliers become zero and

the constraints become inactive, i.e. the initial weights suffice to meet the rate requirements, the

particular user’s rate begins to increase (see e.g. user 2 (green triangles) at∼ 13 dB.).

3.5. Summary 47

4 6 8 10 12 140

0.5

1

1.5

2

2.5

3

SNR [dB]

R [b

ps/H

z] infeasible

4 6 8 10 12 140.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

SNR [dB]

µ

infeasible

change ofπ(·)

Figure 3.9:R andµ = µ + µ′ overS NRfor the same parameters as in Figure3.5

3.5 Summary

In this chapter we presented solutions for the three reference problems for parallel Gaussian

single antenna broadcast and multiple-access channels. Two key properties allow for appealing

algorithms: First, the system can be decomposed to a set of parallel scalar Gaussian channels,

coupled only by the rate and power constraints. This fact makes it possible to use properties of

scalar channels, which in turn are degraded and well understood. Second, there exists a bijective

mapping between rates and powers, which is exploited in the MAC formulation. This property

allows to reformulate the optimization problems in terms ofrates in convex form.

The rate water-filling formulation in the MAC domain is not only appealing as a concrete

interpretation; it has the advantage that rate constraintscan be taken into account explicitly. As

a consequence, the power minimization problem can be solvedmuch more efficiently than in

the BC domain using Algorithm4. In contrast, Algorithm2 in the BC domain has to adjust


the Lagrangian multipliers corresponding to the rate constraints successively while after each

adjustment the resource allocation has to be evaluated. Concerning Problem1 the solution

in the BC domain is more efficient, since the search for the optimal Lagrangian multiplier

corresponding to the power constraint can be done without explicitly evaluating the resource

allocation in each step. This is an advantage compared to Algorithm 5, which solves for the

optimum rate allocation during each step of the bisection process. However, as minimum rate

requirements come into play Algorithm5 and thus the rate water-filling formulation is much

more efficient again, since the constraints are kept implicit duringthe water-filling procedure.

Interestingly, it is not necessary to use different successive decoding orders on different

subcarriers in the MAC as well as the downlink scenario. Instead a single decoding order can

be applied on all subcarriers. The optimum power allocationachieving capacity is such that

users whose Lagrangian weight factors are not ordered according to the ordering of the channel

gains on channelk do not get any power on thekth channel. Moreover the optimum decoding

order changes over the transmit SNR.

Chapter 4

Delay-Limited Transmission over OFDM

Broadcast Channels

In the previous chapter the optimal transmit strategy within a single fading block under various

constraints was studied for single antenna multi-user OFDMsystems. In this setting the fad-

ing process did not play any role and the derived results holdfor arbitrary but time-invariant

channels. Taking into account QoS-constraints from higherlayers in form of rate constraints,

we considered the problem of finding the minimum power which allows to support a given set

of rates. Now, since the wireless channel fades over time, anobvious enhancement is to ask

which sets of rates are supportableindependent ofthe current block fading realization. This

means that the mutual information between transmitter and receivers is kept constant avoiding

fluctuations caused by fades, while the instantaneous powerconstraint is replaced by a long

term power constraint. As a result the probability of erroneous decoding becomes independent

of the fading process.

It is known that in generalmultiple degrees of freedomin fading channels allow reliable

communication independent of the block fading state if the peak power is not bounded. This is

due to the possibility of recovering the information from several independently faded copies of

the transmitted signal. The rate achievable in each fading state is calledzero outage capacity

or alternativelydelay-limited capacity(DLC) [58]. Pioneering work on this topic was carried

out in [65] where the general single user outage capacity was investigated of which the DLC

is a special case. In general, a non-zero DLC exists only if the channel is invertible with finite

power [66]. For single antenna Rayleigh flat fading channels the necessary power is not finite

and thus the DLC is zero. In contrast, multiple antenna systems as well as frequency selective

multi-path channels offer multiple degrees of freedom [67] and thus a non-zero DLC. The topic

of delay-limited transmission has attracted interest recently. Hanly and Tse investigated the case

of flat MACs [58] while Li and Goldsmith addressed the case of flat BCs in [68]. In [69, 70] the

DLC of point to point flat MIMO systems and the influence of transmit correlation on the DLC

was investigated. The case of independently fading parallel channels was studied subsequently

in [71]. A first approach towards frequency selective BCs was made in [72] where the authors

49

50 Chapter 4. Delay-Limited Transmission over OFDM Broadcast Channels

derived a lower bound on the OFDM broadcast DLC region under the restrictive assumption

that equal power is allocated to all subcarriers. Very recently, Caire et al. investigated delay-

limited transmission for the uplink and downlink of cellular systems with many users deriving

large system asymptotics in [61].

In this chapter we focus on the DLC of OFDM downlink systems. We begin with an analysis

of the single user case. This scenario is adequate to characterize the influence of system param-

eters such as the fading distribution and the delay spread onthe DLC. We then turn towards

the broadcast scenario and present an algorithm to evaluatethe OFDM BC DLC-region up to

any finite accuracy. The optimum transmission strategy involves superposition coding in each

fading state which is very complex from a signal processing perspective. Thus in a third step

we turn our attention to orthogonal frequency division multiple access (OFDMA)-transmission

policies. By OFDMA we mean an exclusive assignment of subcarriers so that users are or-

thogonal and no interference handling is needed. Note that we use equivalently OFDMA and

FDMA in the following to indicate exclusive subcarrier assignment. Parts of this chapter have

been also published in [6, 7, 8].

4.1 Single User OFDM Delay-Limited Capacity

In this section we study the single user OFDM DLC and derive some basic results character-

izing the influence of the fading distribution on the DLC. These serve as a background for the

subsequent consideration of the OFDM BC DLC in Section4.2.

4.1.1 System Model

We re-state the system model for the single user case. Consider a point-to-point frequency

selective block-fading channel where OFDM withK subcarriers is applied in order to combat

the intersymbol interference. In analogy to Section2.2 the signal received on subcarrierk is

given by

yk = hkxk + nk ∀ k ∈ K

wherexk ∈ C is the signal transmitted on subcarrierk, nk ∼ CN(0, 1) is circular symmetric

AWGN andK = {1, ...,K}. The sampled frequency response is given by means of the FFT by

hk =

L∑

l=1

cl exp

(

− j2π(l − 1)(k − 1)K

)

∀ k ∈ K (4.1)

whereL ≤ K is the delay spread andcl are the complex path gains that are modeled as indepen-

dent, zero mean random variables with varianceσl > 0 for all l. The vectorσ = [σ1, ..., σL]T is

called thepower delay profile (PDP)and the channel energy is normalized‖σ‖1 = 1. We say

that the channel has a uniform PDP ifσ1 = . . . = σL and a non-uniform PDP otherwise. We

4.1. Single User OFDM Delay-Limited Capacity 51

assume that the PDP is uniform throughout the remainder. Results about the influence of the

PDP are very hard to obtain [8]. The vector of channel gains is given byg ∈ RK+ with gk = |hk|2

and the vector of time domain path gains is denoted byg ∈ RL+ with gl = |cl |2. The distribution of

the path gains is called the (joint) fading distribution. Incase of complex Gaussian distributed

path gains, i.e.ck ∼ CN (0, 1/L) the channel gains follow an exponential distribution with

Pr{gk > x} = e−x. This case corresponds to Rayleigh fading.

Assuming that the fading realizations are known perfectly the system is allowed to allo-

cate power adaptively within each fading block according tothe current channel gainsg =

[g1, ..., gK]T so that

p(g) = [p1(g), ..., pK(g)]T

denotes the power allocation during a fading state with channel gainsg. Giveng the rate achiev-

able over allK parallel subcarriers with a certain power allocationp(g) reads as

R(g, p(g)) =1K

K∑

k=1

Rk (g, pk(g))

=1K

K∑

k=1

log(1+ pk(g)gk) ,

whereRk(g, pk(g)) denotes the rate achievable on subcarrierk with channel gainsg using power

pk(g). As in the previous chapter we introduced the factor 1/K so that all rates are normalized

to spectral efficiency and given in [bps/Hz]. But in contrast to the previous chapter it is assumed

that the system is subjected to a long term power constraint,i.e.

E

1K

K∑

k=1

pk(g)

≤ P,

where the expectation is with respect to the fading process.This means that while the transmitter

is not limited in terms of peak power per fading state, an average power budgetP has to be met.

4.1.2 Rate Water-Filling and Single User OFDM DLC

First we introduce the DLC for a single user OFDM system.

Definition 4.1. The delay-limited capacityCdl of an OFDM system under a long term power

constraintP is given by

Cdl(P)def= sup

Pinfg∈H

R(g,P(g))

whereH is the support of the fading distribution andP : RK 7→ RK is any power allocation


policy advising a power allocationp(g) to every fading stateg ∈ H such that

E

1K

K∑

k=1

pk(g)

≤ P.

In other words,Cdl is the maximum rate which can be achieved in each fading statewith-

out violating the average power constraintP. This is the common definition of delay-limited

capacity, see e.g. [58, 67].

We re-derive the DLC of a single user OFDM system now using theprinciple ofrate water-

filling as introduced in Section3.3.1 in the context of multi-user systems, which leads to an

interesting perspective. Definition4.1already implies that an optimum power allocation policy

P has to be such that in each fading state the power allocationp(g) that supports the given rate

Cdl with minimum power is found. For anyg ∈ H this optimization problem is equivalent to

minK∑

k=1

pk(g) − µK∑

k=1

Rk(g, pk(g))

where the dual parameterµ ≥ 0 has to be chosen such that the rate constraint

1K

K∑

k=1

Rk(g, pk(h) ≥ Cdl

is met. Using the relation between power and rateRk(g, pk(g)) = log(1+ pk(g)gk) on subcarrier

k and dropping the argumentg this problem can be expressed as

minK∑

k=1

(

eRk − 1gk

− µRk

)

.

The resulting optimality conditions are given by

Rk =

[

log(µ) − log

(

1gk

)]+

∀k ∈ K , (4.2)

Cdl =1K

K∑

k=1

Rk. (4.3)

Obviously (4.2) and (4.3) reflect just the single user version of the optimality conditions (3.31)

in Section3.3.1. Combining (4.2) and (4.3) for all K subcarriers and solving for the Lagrangian

multiplier µ yields

µ =exp

(CdlK

dg

)

∏

k∈D(g) gd−1

g

k

, (4.4)

where the random variableD (g) ⊆ K denotes the set ofactivesubcarriers anddg = |D (g)|is the cardinality of this set. Note that from (4.2) follows by simple algebra that the allocated


power reads aspk = µ − g−1k for anyk ∈ D (g) and zero otherwise. Substituting the Lagrangian

multiplier given by (4.4) in the average power expression

P = E

1K

K∑

k=1

pk(µ)

= E

1K

K∑

k=1

[

µ − 1gk

]+

yields an implicit characterization for the single user OFDM delay-limited capacityCdl with

average power constraintP:

P = E

dg exp(

CdlKdg

)

K∏

k∈D(g) gd−1

g

k

− 1KE

∑

k∈D(g)

1gk

(4.5)

Since the numerator of the first term in (4.5) can be lower bounded by a constant the delay-

limited capacityCdl is greater than zero if

∫

RK+

1∏

k∈D(g) gd−1

g

k

dFg (g) < ∞. (4.6)

Here,Fg denotes the joint fading distribution function of channel gains. The class of fading

distributions for which (4.6) holds withD (g) ≡ K is calledregular in [67].

Note that it is extremely difficult to characterize this expression for arbitraryL < K since

due to the oversampling the channel gainsgk are highly correlated and the fading process in

the frequency domain is characterized by a lower-dimensional distribution [8]. Thus we mainly

focus on the caseL = K in the following.

4.1.3 Low SNR Behavior

In this section we study the behavior of the OFDM DLC at low SNR. First of all we characterize

the first and second order behavior, subsequently we conclude under which conditions delay-

limited transmission with non-zero rate is possible.

The first order behavior is quite meaningful from a physical point of view. It reveals the

minimum energy per bit at which reliable communication is possible under delay limitations.

The following consideration yields a good intuition: The energy per transmitted bit is given by

Eb =P

Cdl(P). (4.7)

Since the thermal noise variance is normalized, this expression gives theEb/N0 as a function

of the transmit powerP. Obviously (4.7) is monotonically increasing inP. Thus limP→0 Eb(P)

yields the minimumEb/N0, where the reciprocal is exactly the first order behavior as the SNR

tends to zero. Note that in the present context the bandwidthis fixed and thus it is rather a low

SNR than a wideband limit implying that the rate tends to zeroas the limit is approached.


In order to characterize how the spectral efficiency behaves overEb/N0, Verdu introduced

the notion ofwideband slopein [73]. If we want to determine this quantity for delay-limited

transmission, the second order behavior over SNR is needed.Interestingly, the multiplicity

of the maximum subcarrier gain occurs in the expression. This is of relevance if the fading

distribution has point masses. For example limited feedback and thus a finite resolution of the

channel leads to this scenario. The following theorems characterize the first and second order

behavior. For the ease of notation we define the maximum channel gaing∞def= ‖g‖∞.

Theorem 4.2. If E{

g−1∞

}

< ∞ then the 1st order limit of the OFDM DLC is given by

C′dl (0)def= lim

P→0

Cdl

(

P)

P=

1E

{

g−1∞

}

(

nats/Hzws

)

. (4.8)

The minimumEb

N0, at which reliable delay-limited communication is possible, is thus given by

(

Eb

N0

)

min

= log(2)E{

g−1∞

}

. (4.9)

The expression in (4.9) characterizes the minimumEb

N0, at which a non-zero spectral effi-

ciency can be achieved for the considered system with fixed bandwith. So (4.9) is not a wide-

band limit in the classical sense. A direct consequence of Theorem4.2 is that in the low SNR

regime it is optimal to serve only the best subcarrier.

Proof. We begin with the implicit expression for the OFDM DLC given in (4.5). The power

series exp(x) =∑∞

n=0xn

n! yields exp(x) ≥ 1+ x and thus a lower bound on the required powerP.

P ≥ E

dg

(

1+ CdlKdg

)

K∏

k∈D(g) gd−1

g

k

− 1KE

∑

k∈D(g)

1gk

(4.10)

Now assume that all subcarriers have maximum channel gain, i.e.gk = ||g||∞ = g∞ for all k ∈ Kand thusdg = K. Clearly, this scenario yields a lower bound on the necessary sum power with

P ≥ E{

1+Cdl

g∞

}

− E{

1g∞

}

= CdlE

{

1g∞

}

. (4.11)

Hence we obtain an upper bound onCdl which reads as

Cdl ≤P

E{

g−1∞

} . (4.12)

We now derive an upper bound on the required powerP and thus a lower bound onCdl. To

this end we fixǫ > 0 and set the number of supported subcarriers to one, i.e.dg = 1, supporting


exclusively one of the subcarriers with maximum channel gain. Clearly, this is a suboptimal

strategy. Since exp(x) ≤ (1+ ǫ)x for sufficiently smallx and anyǫ we get for sufficiently small

P

P ≤ E{

1+ (1+ ǫ)CdlKKg∞

}

− 1KE

{

1g∞

}

.

Hence we have

Cdl ≥P

(1+ ǫ)E{

g−1∞

} (4.13)

for anyǫ > 0. Combining the upper bound from (4.12) and the lower bound from (4.13) yields

the desired result. �

Theorem4.2characterizes the first order behavior of the DLC at low SNR and thus the min-

imum Eb

N0at which delay-limited transmission is possible. To get a more precise characterization

of the DLC and to determine an equivalent to the wideband-slope, the second order behavior is

of interest. We have the following result.

Theorem 4.3. Define the sub-linear term∆dl

(

P)

= C′dl (0) P− Cdl

(

P)

. If E{

g−1∞

}

< ∞ then the

2nd order limit of the OFDM DLC is given by

C′′dl(0)def= lim

P→0

∆dl

(

P)

P2=

KE{

χ−1g g−1

∞}

[

E{

g−1∞

}]3(4.14)

whereχg is a random variable representing the multiplicity of subcarriers with maximum chan-

nel gain. For absolute continuous joint fading distributions without point masses(4.14) simpli-

fies to

C′′dl(0)def= lim

P→0

∆dl

(

P)

P2=

K[

E{

g−1∞

}]2. (4.15)

Proof. The starting point is again (4.5) and we use the power series exp(x) =∑∞

n=0xn

n! up to the

quadratic term yielding exp(x) ≥ 1 + x + 12x2. We then choose the following strategy. We fix

ǫ > 0 and for all subcarriersk with gk ≥ g∞ − ǫ (i.e. subcarriers with channel gain close to the

maximum channel gain) we set the channel gains also tog∞. The variableχg (ǫ) denotes the

number of subcarriers having maximum channel gain then. Since channel gains are increased,

this clearly yield a lower bound on the necessary power. For sufficiently smallP follows

P ≥E

χg (ǫ)(

1+ CdlKχg(ǫ) +

C2dlK

2

2χ2g(ǫ)

)

Kg∞

− 1KE

{χg (ǫ)

g∞

}

=E

Cdl +12Kχ−1

g (ǫ) C2dl

g∞

(4.16)

=E{

g−1∞

}

Cdl +12

KE{

g−1∞ χ

−1g (ǫ)

}

C2dl. (4.17)


The expression in (4.17) describes an upward open parabola inCdl. Solving this equation for

Cdl yields for the positive root the following inequality.

Cdl

(

P)

≤ −E

{

g−1∞

}

KE{

χ−1g (ǫ) g−1

∞}

+

√√

E2{

g−1∞

}

K2E2{

χ−1g (ǫ) g−1

∞} +

2P

KE{

χ−1g (ǫ) g−1

∞}

(4.18)

Using the Taylor expansion of the square root at zero yields the following upper bound for any

fixed ǫ′ > 0 and sufficiently smallx:

√ax+ b ≤

√b+

12

a√

bx− 1

4(1+ ǫ′)a2

b3/2x2. (4.19)

Using (4.19) we can rewrite (4.18) as

Cdl

(

P)

≤ 1E

{

g−1∞

} P−KE

{

χ−1g (ǫ) g−1

∞}

(1+ ǫ′)E3{

g−1∞

} P2. (4.20)

Subtracting the first order expression (4.8) from (4.20) we arrive for someǫ′′(ǫ) > 0 at

∆dl

(

P)

≥KE

{

χ−1g (ǫ) g−1

∞}

(1+ ǫ′)E3{

g−1∞

} P2

≥KE

{

χ−1g g−1

∞}

− ǫ′′

(1+ ǫ′)E3{

g−1∞

} P2 (4.21)

and thus have established a lower bound on∆dl(P) for anyǫ, ǫ′ > 0. The last inequality (4.21)

follows from the following argument: Observe thatχg (ǫ) ≥ 1. Further for any channel gain

realizationg we have

limǫ→0

χ−1g (ǫ) g−1

∞ = χ−1g g−1

∞

almost surely with respect to the fading distribution. Provided thatE{

χ−1g g−1

∞}

≤ E{

g−1∞

}

< ∞we obtain by the dominated convergence theorem [74]

limǫ→0E

{

χ−1g (ǫ)g−1

∞}

= E{

χ−1g g−1

∞}

leading to (4.21).

In analogy to the derivation of the lower bound we can derive

∆dl

(

P)

≤(1+ ǫ′′′) KE

{

χ−1g g−1

∞}

E3{

g−1∞

} P2. (4.22)

for any ǫ′′′ > 0. Combining (4.21) and (4.22) leads to the desired result. The simplification


for absolute continuous fading distributions in (4.15) follows from the fact that in this case the

event of more than one subcarrier having the maximum channelgain has measure zero. �

Using the previous two theorems we get the following characterization of the slope ofdelay-

limited spectral efficiencyover Eb

N0, in analogy to the wideband-slope in [73]. Again, note that

wideband-slopeis a misleading term in the present context since the bandwidth is fixed.

Corollary 4.4. For absolute continuous fading distributions, the slope ofthe delay-limited spec-

tral efficiency overEbN0

is given by

Sdl(0) =2(C′dl)

2

C′′dl

=2K

(b/s/Hz/(3dB)). (4.23)

This follows from [73, Thm. 9].

Surprisingly the number of subcarriersK occurs in the denominator in (4.23). An intuitive

explanation is that only 1/K of the spectrum has the maximum channel gain. While this is

irrelevant for(

Eb

N0

)

min, where only the valueg∞ is of interest, the slope is quite sensitive to the

fraction of the spectrum1K . If fading processes including point masses, the expected multiplicity

of the maximum channel gain occurs additionally increasingthe fraction of the spectrum.

We now characterize under which conditions on the joint fading distribution delay-limited

transmission with a non-zero rate is possible. In order to doso we recall that a prerequisite for

the previous results was that the expectation of the inversemaximum channel gain is bounded,

i.e. E{

g−1∞

}

< ∞.

Theorem 4.5.Suppose that at least two path gains have a joint fading distribution with bounded

density in some open neighborhood of zero. Then, zero outagetransmission is possible under

an average power constraint.

Proof. We examine under which conditionsE{

g−1∞

}

< ∞ holds. Applying the inequalityE {|X|} ≤1 +

∑∞n=1 Pr{|X| ≥ n} for some real random variableX from [75, p.17] the expectation can be

written as

E{

g−1∞

}

≤ 1++∞∑

n=1

Pr

{

g∞ ≤1n

}

≤ 1++∞∑

n=1

Pr

{

||g||1 ≤1n

}

≤ 1++∞∑

n=1

Pr

{

||g||∞ ≤1n

}

≤ 1++∞∑

n=1

Pr

L⋂

l=1

{

gl ≤1n

}

(4.24)

where we used the fact that‖g‖∞ ≤ ‖g‖1 = 1K ‖g‖1 ≤ g∞.


By assumption there are two path gains ˜g1, g2 with a joint fading distribution and density

bounded by some real constant 0≤ c0 < ∞ in some open neighborhood of zero, where the

indices 1 and 2 are chosen without loss of generality. Thus there existsn0 such that the density

is bounded byc0 on [0, 1/n0] × [0, 1/n0]. Hence, we have forn ≥ n0

Pr

L⋂

l=1

{

gl ≤1n

}

≤

∫

[0,1/n]×[0,1/n]

dF (g1, g2)

≤ c0

∫

[0,1/n]×[0,1/n]

dg1dg2

≤ c0

n2

rendering the sum finite. As a consequence the DLC must be non-zero. �

Theorem4.5 gives a sufficient condition for regularity of as defined in [67]. It makes the

notion of multiple degrees of freedommore explicit: Any frequency selective channel with at

leat two taps fulfilling the assumptions in Theorem4.5allows for delay-limited transmission. It

remains to be stressed that the taps do not have to be independently distributed. They can even

be highly correlated.

The behavior of the OFDM DLC overEb

N0and the first and second order approximations

in the low SNR regime characterized in Theorems4.2 and4.3 are illustrated in Figure4.1 for

different numbers of i.i.d. complex Gaussian taps andL = K. It can be observed that the

approximation is very tight for smallL while it degrades asL increases or equivalently, the

range shrinks for which the approximation is tight. This is due to the fact that for largeL the

different channel gains change the slope quite heavily until allsubcarriers are served in average.

Figure4.2 illustrates the OFDM DLC overEb/N0 for an increasing number of i.i.d. complex

Gaussian taps andL = K. For the purpose of comparison the capacity of a Gaussian channel

is depicted as a red dashed line which yields the well-known (Eb/N0)min = −1.59dB. As L

increases reliable delay-limited communication is possible at smaller and smaller (Eb/N0)min,

while at the same time the slope decreases . For the case of i.i.d. complex Gaussian taps and

L sufficiently large (Eb/N0)min scales with 1log(L) so that (Eb/N0)min is not bounded from below.

This behavior is intuitive, since at low SNR only the best subcarrier is served. On the other hand

the subcarrier with maximum channel gain occupies only 1/K of the entire system bandwidth,

which explains the decrease ofSdl(0) asK increases. This general behavior holds for a wide

class of fading distributions since due to results from order statics the scaling of the maximum

channel mainly depends on the number of taps. This means thatthe delay spread, i.e. the

number of paths, governs the behavior of the OFDM DLC at low SNR.


−4 −3.5 −3 −2.5 −2 −1.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

−3

Eb/N

0 [dB]

C d

l [bps

/Hz]

Cdl

1st order2nd order

−7 −6.5 −6 −5.5 −5 −4.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

−3

Eb/N

0 [dB]

C d

l [bps

/Hz]

C dl

1st order 2nd order

−9 −8.5 −8 −7.5 −7 −6.5 −60

0.5

1

1.5

2

2.5

3

3.5

4

x 10−3

Eb/N

0 [dB]

C d

l [bps

/Hz]

C dl

1st order2nd order

−10.5 −10 −9.5 −9 −8.5 −8 −7.50

1

2

3

4

5

6

7

8

x 10−3

Eb/N

0 [dB]

C d

l [bps

/Hz]

Cdl

1st order2nd order

Figure 4.1: OFDM DLC, 1st and 2nd order behavior overEb/N0 at low SNR forL = K andL = 4 (upper left)L = 16 (upper right),L = 64 (lower left) andL = 512 (lower right)

−12 −10 −8 −6 −4 −2 0 2

1

2

3

4

5

6

7

8

9

x 10−3

Eb/N

0 [dB]

C d

l [bps

/Hz]

L = 2, ..., 1024

Figure 4.2: OFDM DLC, forL = K = 2N, N = 1, ..., 10 at low SNR. The red dashed curverepresents the Gaussian channel with

(Eb

N0

)

min= −1.59dB


4.1.4 High SNR Behavior

After studying the OFDM DLC at low SNR we turn towards the highSNR regime now. An

important result was derived in [67, page 1279]. It was shown that a lower bound on the DLC

for regular fading distributions is given by

Cdl

(

P)

≥ log

(

PE {g}

)

, (4.25)

where the quantity ¯gdef=

∏Kk=1 g−1/K

k is the geometric mean of the inverse channel gains. The

authors showed that allocating equal power

pk =exp(Cdl)

g(4.26)

to all subcarriersk ∈ K achievesCdl with an average powerE{∑Kk=1 pk} ≥ P . Substituting (4.26)

into the average power inequality then leads to the lower bound in (4.25).

We can extend this result and establish a similar upper bound.

Lemma 4.6. For regular fading, i.e.E {g} < ∞, and for sufficiently largeP the OFDM DLC is

upper bounded by

Cdl

(

P)

≤ log

P(

1+ 1K

)

E {g}

. (4.27)

Proof. The upper bound onCdl can be shown using the following strategy: For any fading state

and any subcarrier we assign a minimum channel gain ofǫ > 0 in order to avoid zero channels.

Clearly, since channel gains are only increased by this assignment this yields an upper bound.

Definegǫkdef= max{gk, ǫ}. Then, using (4.5) we have for sufficiently largeP

P ≥ E

exp(Cdl)

K∏

k=1

(

gǫk)− 1

K

− 1

K

K∑

k=1

E

{

1gǫk

}

(4.28)

= E

exp(Cdl)

K∏

k=1

(

gǫk)− 1

K

− E

{

1gǫ1

}

.

The second term grows without bound asǫ goes to zero for many fading distributions including

Rayleigh fading. This term can be bounded as follows.

E

{

1gǫ1

}

=

∫ ∞

0

1gǫ1

dFg (g1)

≤ ǫ−1. (4.29)

Clearly,ǫ is related toP, since in (4.28) it is assumed that all subcarriers are supported. Since

the minimum channel gain is at leastǫ and the underlying optimal power control law is water-


filling the inequality (4.28) holds if

P ≥ Kǫ

which is very loose. Hence, with (4.29) we obtain

E

{

1gǫ1

}

≤ PK

and finally for anyǫ > 0

Cdl ≤ log

P(

1+ 1K

)

E

{∏K

k=1

(

gǫk)− 1

K}

.

Furthermore we haveK∏

k=1

(

gǫk)− 1

K ≤ g

for any fading realization with the limit

limǫ→0

K∏

k=1

(

gǫk)− 1

K = g.

Hence, by dominated convergence we obtain

limǫ→0E

K∏

k=1

(

gǫk)− 1

K

= E {g}

provided that the expectationE {g} < ∞ exists which is true by the assumption of regular fading.

Thus we obtain (4.27). �

As a consequence of Lemma4.6the OFDM DLC at sufficiently high SNR lies in a corridor

becoming tight for largeK.

log(

P)

+ log

(

1E {g}

)

≤ Cdl

(

P)

≤ log

(

P

(

1+1K

))

+ log

(

1E {g}

)

This target corridor is basically determined by the additive expression log(

1E{g}

)

depending on

the fading distribution. This expression characterizes the additive offset compared to the Gaus-

sian channel capacity scaling log(P) at high SNR. We can derive a simple bound on this quantity.

Lemma 4.7. If the fading process is regular, i.e.E {g} < ∞, then the asymptotic offset is lower

bounded by

log

(

1E {g}

)

≤ E {

log(g1)}

.


Proof. Using (4.27) and setting

E {g} = E {

exp(

log(g))}

we get by Jensen’s inequality

log

(

1E {g}

)

= − logE{

exp(

log(g))}

≤ − log(

exp(

E{

log(g)}))

(4.30)

= −E {

log(g)}

= −E

− 1

K

K∑

k=1

log(gk)

=

∫ ∞

0log(g) dFg1 (g) ,

whereFg1 is the cdf of the channel gain on the first (or any other) subcarrier. This yields the

desired result. �

As a consequence we have the following upper bound on the OFDMDLC.

Lemma 4.8. If the fading process is regular, i.e.E {g} < ∞, for sufficiently largeP the OFDM

DLC is upper bounded by

Cdl(P) ≤ log(P) +1K+ E{log(g1)}. (4.31)

Further define g(K)def= 1

K

∑Kk=1 log(g−1

k ) and assume that g(K) → E{log(g−11 )} in probability. If

exp(g(K)) is uniformly integrable then the bound becomes asymptotically tight as K→∞.

Proof. The bound follows from Lemma4.7 and the fact that log(x) ≤ x − 1. Using uniform

integrability and convergence in probability it further follows from [76, Thm. 3.5.] that the

bound becomes tight. �

Note that the bound holds for arbitraryL andK and independent of the relation betweenL

andK. Moreover, forL = K with i.i.d. taps and assuming thatg(K) → E{log(g−11 )} the bound

becomes tight. Obviously a pure oversampling of the spectrum does not suffice to assure the

convergence in probability. In [9] it was further shown that under the assumptions in the second

part of Lemma4.8the upper bound is also a tight upper bound for the ergodic capacity and thus

the OFDM DLC converges to ergodic capacity. This is consistent with the results from [67].

Figure4.3 illustrates the bound forL = K and the simulated curves for i.i.d. Rayleigh fading

with increasingL = K. It can be observed that the bound is approached quite quickly.

To summarize, it turns out that at high SNR the expressionE{

log(g1)}

is of central impor-

tance and thus the fadingdistributiongoverns the behavior of the OFDM DLC. In contrast, the

4.2. Multi-User OFDM Delay-Limited Capacity 63

10 15 20 25 30 35 40 452

4

6

8

10

12

14

SNR [dB]

C d

l [bps

/Hz]

log(P) + E{log(g1)}

L = 2, ..., 32

Figure 4.3: Illustration of the OFDM DLC at high SNR for i.i.d. complex Gaussian taps withL = K = 2N, N = 1, ..., 5 and the upper bound log(P) + E

{

log(g1)}

number of tapsL or equivalently the delay spread is predominant in the low SNR regime. With

these main results we can turn towards the multi-user case.

4.2 Multi-User OFDM Delay-Limited Capacity

For the case of point to point-transmission it was possible to characterize the OFDM DLC using

an implicit expression. Turning towards the multi-user case this implicit expression becomes

more complicated. However, if the fading distributions of all users are regular, a non-empty set

of rates is achievable under a common long term power constraint. This leads to the notion of a

delay-limited capacity region. We will focus on the downlink case in the following. The uplink

can be treated in an analog manner with the only difference that a set ofM power constraints

has to be taken into account.

4.2.1 The OFDM BC Delay-Limited Capacity Region

Let us introduce the OFDM BC DLC regionCdl

(

P)

. Generalizing the characterization of the

single user OFDM DLC given in Definition4.1 to a broadcast scenario withM users leads to

the subsequent statement.

Definition 4.9. A rate vectorR lies in the DLC regionCdl

(

P)

with a long term sum power

constraintP if and only if for any fading stateg there is a powerP′ being a solution to

min P s.t. R ∈ CBC (g,P) (4.32)


Algorithm 6 OFDM DLC Region Evaluation(1) Determine for each userm ∈ M its single user DLC (thus the intersection of the DLCregion with the own axis)while desired accuracy not reacheddo

(2) For any two neighboring vectorsR1 ∈ Bdl

(

P)

and R2 ∈ Bdl

(

P)

on the boundarycalculate interpolated vectorRint = 1/2(R1 + R2)(3) Adjustα > 1 by bisection using Alg.4 from Section3.3.1such thatαRint ∈ BDL

(

P)

end while

with CBC (g,P) being the OFDM broadcast channel capacity region for fixed channel gains

given in (2.23) and

E {P′} ≤ P. (4.33)

Furthermore,R lies on the boundaryBdl

(

P)

if and only if

E {P′} = P.

Note that as in the single user case we choose to state the DLC region as a definition rather

than a theorem as done in [58]. There it is shown that this definition coincides with the set of

rates that can be achieved under a long term power constraintP with decoding error probability

only depending on the decoding delay and becoming arbitrarysmall as the codelength goes to

infinity, if the fading process is jointly stationary ergodic and the codewords can be adapted to

the realization of this fading process.

4.2.2 An Algorithm to Evaluate the OFDM BC DLC Region

To evaluate the OFDM delay-limited capacity region turns out to be difficult. This is due to

the fact that Definition4.9yields only animplicit characterization of the DLC region for a fixed

sum powerP. It means that similar to the single user case we can check forevery rate vector

R whether it can be supported or not simply by checking the average power condition in (4.33)

whereP′ is a random variable. Unfortunately it is not possible to express the pdf ofP′ in closed

form. HoweverE {P′} can be approximated solving

min P s.t. R ∈ CBC (g,P) (4.34)

for a sufficiently large number of Monte Carlo runs and thus fading realizationsg. In order to

solve (4.34) for a single fading state, the problem can be considered in the dual uplink and the

iterative rate water-filling algorithm given as Alg.4 in Chapter3 can be used. Alternatively,

Alg. 2 from Section3.2.3can be chosen solving the problem directly in the broadcast setting.

In order to evaluate the OFDM BC DLC regionCdl

(

P)

one can proceed as follows. First the

single user DLC ratesCmdl

(

P)

have to be calculated for all usersm ∈ M. This is done by single

user rate water-filling over Monte Carlo runs or equivalently using Alg.4, where the number of

4.3. Delay-Limited Transmission under FDMA Constraints 65

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5

3

3.5

R1[bps/Hz]

R2[b

ps/

Hz]

R(n)int (α = 1)

R(n+1)(α = 1.08)

Figure 4.4: Example for an iteration of the described algorithm to calculate the OFDM BC DLCregion forM = 2 users withL = 7 i.i.d taps each andK = 16 subcarriers at 10dB

users is set toM = 1. Since the characterization is only implicit bisection can be used to find

the optimum single user rate becauseCmdl

(

P)

is monotone inP.

Due to the convexity ofCdl

(

P)

any convex combinationRint must lie insideCdl

(

P)

. On the

other hand, the single user ratesRm are a component-wise upper bound for all other rate vectors.

Since the necessary powerP (αRint) is monotone inα, simple bisection can determine a rate

tuple on the boundary of the region for each angle and thus foreach ray through the origin (see

Fig. 4.4). For any new point on the boundary the refinement procedure can be repeated until the

desired number of points defining the border is obtained. This procedure is summarized in Alg.

6. Note that if the fading distribution is the same for all users the region is symmetric and can

be constructed by mirroring one calculated sector. This reduces the computational complexity

significantly. An exemplary step of the algorithm is depicted in Figure4.4.

The drawback of Alg.6 is its high complexity. It involves not only multiple nestedbisec-

tion processes. Each single step also consists of the evaluation of Monte-Carlo runs causing a

prohibitive computational cost.

4.3 Delay-Limited Transmission under FDMA Constraints

The optimum transmission scheme to achieve a certain rate tuple in general involves superposi-

tion coding and subcarrier-sharing. Superposition codingis a complicated information theoretic

concept relying on a nonlinear successive decoding operation at the receivers. From a signal

processing point of view this is very inexpedient and hardlyrealizable. FDMA transmission

schemes are a suboptimal but attractive alternative especially for real world applications since

the complexity caused by successive decoding can be avoided. Thus we focus on schemes rely-


ing on an exclusive assignment of subcarriers in the following. To begin with, in Section4.3.1

we develop lower bounds on the OFDM BC DLC region which only exploit ordinal channel

state information, i.e., where only the ordering of the channel gains influences the resource al-

location scheme. Subsequently we present conditions for the optimality of FDMA in Section

4.3.2and then study the problem of finding close-to optimal FDMA schemes in Section4.3.3.

4.3.1 Lower Bounds on the OFDM BC DLC Region Based on FDMA

It is evident that a major difficulty in order to evaluate the OFDM BC DLC region is the iterative

rate water-filling procedure which has to performed over a sufficiently large number of channel

realizations in order to represent the fading statistics. The intention of this section is to circum-

vent this problem by using the rate water-filling expressions (4.2) and (4.3) from Section4.1.2

in combination withexpected orderedchannel gains. The idea is to use simply the information

which subcarrier is the best, the second best and so on and to allocate fixed rate budgets to these

ordered subcarriers. This perspective brings into play thenotion of order statistics. Based on

this principle we derive lower bounds on the OFDM BC DLC region.

For the analysis we need the following convention. For a given vectorg ∈ RK let us intro-

duce the total ordering

gk[K] ≥ gk[K−1] ≥ . . . ≥ gk[1] ,

i.e. gk[1] is the minimum value andgk[K] is the maximum value. Ifg is a random vector then

the distribution ofgk[ p] is known to be thep-th order statistic. The p-th order statistic can be

explicitly given for K independent random variablesgk, k = 1, ...,K with distributionFg and

density fg from standard books (see e.g. [77]). The p-th order density is given by

fgk[p] (x) = K fg (x)

(

K − 1p− 1

)(

Fg (x))p−1 (

1− Fg (x))K−p

. (4.35)

Further define the terms

ζpdef=

∫ ∞

0g−1 dFgk[p] (g) (4.36)

representing the expected inverse of thep-th ordered element. Assuming that the channel gains

are i.i.d. (4.36) combined with (4.35) yields the expected noise level of thep-th ordered sub-

carrier. Together with the order information we can now deduce a resource allocation strategy

assigning a fixed rate budget to thep-th ordered subcarrier. By doing so the water-filling pro-

cedure in each fading state can be avoided. The following lemma yields a lower bound on the

OFDM DLC region implying an FDMA strategy.

Lemma 4.10. Let i ∈ [1,M]K be a vector of user indices and letKm(i)def= {k ∈ K : ik = m} be

the set that contains all indices of elements ofi which are equal to m. Further letIs ⊂ [1,M]K


be the subset containing thesei in which all users m occur at least s times, i.e.

Isdef= {i : |Km(i)| ≥ s, ∀ m∈ M} . (4.37)

Then for each choice of s∈ N the average required power P to support any rate vectorR in

each fading state is upper bounded by

P ≤∑

i∈Is

M∑

m=1

|Km(i)|∑

p=1k[ p]∈Km(i)

eRm,k[p] (i) − 1MK ζ−1

m,p

+(

MK − |Is|)

M∑

m=1

⌊K/M⌋∑

p=1

eRm,k[ p] − 1MK θ−1

m,p

(4.38)

with

Rm,k[p] (i) =[

log(λ) − log(ζm,p)]+

(4.39)

whereλ is chosen such that rate constraint for user m is fulfilled

1K

|Km(i)|∑

p=1

Rm,k[p] (i) = Rm

and the expected channel gains are given by

ζm,pdef=

∫ ∞

0

1x

dFg′k[p]

(x) (4.40)

with g′k[ p] being the p-th order statistic of|Km(i)| random variables g′ with cumulative density

function

Fg′ (x) = M∫ x

0Fg(τ)

M−1 fg(τ)dτ. (4.41)

Similarly the rateRm,k[ p] is the solution to

Rm,k[p] =[

log(λ) − log(θm,p)]+

(4.42)

with λ chosen such that1K

⌊K/M⌋∑

p=1

Rm,k[p] = Rm

and the expected channel gains are given by

θm,pdef=

∫ ∞

0

1x

dFg′′k[p]

(x) (4.43)

with g′′k[ p] being the p-th order statistic of

⌊KM

⌋

random variables g′′ with cumulative density

function

Fg′′ (x) = 1− M∫ ∞

xFg(x

′)M−2(

Fg(x′) − Fg(x)

)

fg(x′) dx′. (4.44)


Proof. The basic idea is to distinguish between two cases: The case where each user has the

best channel at least ons subcarriers and the case where at least one user has on less than s of

the K subcarriers the best channel. LetI ≡ [1,M]K denote the set of allMK possible index

combinations. Withi ∈ I we define the event

Hidef=

{

ω : gi1,1 (ω) > gl,1 (ω)∣∣∣l,i1

, . . . , giK ,K > gl,K (ω)∣∣∣l,iK

}

,

i.e., the event where useri1 is best on subcarrier 1, useri2 is best on subcarrier 2 and so on. Note

that the absolute continuity of the fading distribution yields

∑

i∈IPr{Hi} = 1

since the remaining events occur with probability zero. Thus the necessary powerP reads as

P = E {P′} = 1MK

∑

i∈Is

E {P′| Hi} +1

MK

∑

i<Is

E {P′|Hi}

whereIs ⊂ [1,M]K is given in (4.37) and represents the subset containing the elements where

all users occur ini at leasts times. We derive a bound for the first addend first. LetKm(i) be the

set that contains all indices of elements ofi that are equal tom. Fixing i andmand ordering the

values according to

gm,k[|Km(i)|] ≥ . . . ≥ gm,k[1] , k[

p] ∈ Km(i)

the first term on the right hand side is bounded by

∑

i∈Is

E {P′| Hi} ≤∑

i∈Is

M∑

m=1

|K im|∑

p=1k[ p]∈K i

m

E

eRm,k[ p] (i) − 1MK gm,k[p]

∣∣∣∣∣∣∣

Hi

(4.45)

with Rm,k[p] (i) such that the required rates are supported. The expectation on the RHS is inde-

pendent of the actual referred subindex in the multi-indexi and depends only on the number of

entries of userm in i counted by the setKm(i). Thus we can replace the RHS and rewrite the

inequality in (4.45) as

∑

i∈Is

E {P′|Hi} ≤∑

i∈Is

M∑

m=1

|Km(i)|∑

p=1k[ p]∈Km(i)

eRm,k[ p] (i) − 1MK

E

1gm,k[p]

≤∑

i∈Is

M∑

m=1

|Km(i)|∑

p=1

eRm,k[ p] (i) − 1MK ζ−1

m,p

whereζm,p is the expectation of thep-th inverse best channel coefficient given in (4.40). The


cdf Fg′(x) and pdf fg′ (x) of the best channelg′ = maxm gm,k are independent ofk and read as

Fg′ (x) =Pr

{

g1,1 ≤ x, g1,1 > gl,1,l,1}

Pr{

g1,1 > gl,1,l,1}

= M∫ x

0Fg(τ)

M−1 fg(τ) dτ (4.46)

and

fg′ (x) =dFg′ (x)

dx= M Fg(x)M−1 fg(x) (4.47)

yielding the desired result for the first term.

In order to bound the second term a different strategy has to be used. For all cases repre-

sented in the complementary setIs ≡ I \ Is at least one user occurs ini ∈ Is less thans times

and hence has the best channel on less thans subcarriers. For the cases = 1 the strategy for

the previous term can not even guarantee his delay-limited rate requirement since at least one

user does not get any subcarrier. Alternatively, we simply divide the set of subcarriers inM

sub-bands so that⌊K/M⌋ subcarriers are allocated to each user. We then perform ratewater-

filling as done for the first term and take the best out of this set. Hence the second term is upper

bounded by

∑

i∈Is

E {P′| Hi} ≤∑

i∈Is

E

M∑

m=1

⌊K/M⌋∑

p=1

eRm,k[p] − 1gm,k[p]

∣∣∣∣∣∣∣

Hi

≤∑

i∈Is

M∑

m=1

⌊K/M⌋∑

p=1

(

eRm,k[ p] − 1)

E

1gm,k[p]

∣∣∣∣∣∣∣

Hmi

(4.48)

where the eventHmi is defined as

Hmi

def=

{

ω : gn1,1 (ω) > gm,1 (ω)∣∣∣m,n1

, . . . , gnK ,K (ω) > gm,K (ω)∣∣∣m,nK

}

.

The second inequality stems from the fact that the expectation is further conditioned on the

eventHmi assuming that userm does not have the maximum channel gain on any subcarrier.

Since all subcarriers are independent we define for each subcarrier the following conditioned

probability and get after some manipulations

Fg′′ (x) = 1− Pr{

gm,k > x∣∣∣ gn,k > gm,k, n , m

}

= 1− Pr{

g1,1 > x, g2,1 > gl,1,l,2}

Pr{

g2,1 > gl,1,l,2}

= 1− Pr{

x < g1,1 < g2,1, gl,1,l,1,2 < g2,1}

Pr{

g2,1 > gl,1,l,2}

= 1− M∫ ∞

x

(

Fg(x′) − Fg(x)

)

Fg(x′)M−2 fg

(

x′)

dx′ (4.49)


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

R1 [bps/Hz]

R2 [b

ps/H

z]

FDMA−bound (s=1)FDMA−bound (s=3)FDMA−bound (s=4)OFDM BC DLC region

Figure 4.5: OFDM BC DLC region for 16 subcarriers at -20 dB. The lower bound is shown fordifferent values of the parameters

with pdf

fg′′ (x) =dFg′′ (x)

dx. (4.50)

Thus, with (4.35) we can express the conditioned expectation as

E

1gm,k[p]

∣∣∣∣∣∣∣

Hmi

=

∫ ∞

0

1x

dFg′′k[p]

(x)

leading to (4.43). Since the addends do not depend on the indexi the first sum in (4.48) can be

substituted by the factor∣∣∣Is

∣∣∣ = MK − |Is| leading to

∑

i∈Is

E {P′|Hi) ≤

(

MK − |Is|)

M∑

m=1

⌊K/M⌋∑

p=1

(

eRm,k[p] − 1)

MK θ−1m,p

.

This concludes the proof. �

Corollary 4.11. For the case of i.i.d. Rayleigh fading, i.e. f(x) = e−x, with L = K the expression

in (4.41) simplifies to

Fg′(x) =(

1− e−x)M (4.51)

and (4.44) yields

Fg′′(x) = 1− M∫ ∞

x

(

1− e−x′)M−2 (

e−x − e−x′)

e−x′ dx′. (4.52)

Lemma4.10 yields a bound which depends ons. This parameter reflects the number of

subcarriers on which each user has the best channel and thus changes the relation between term


0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

R1 [bps/Hz]

R2 [b

ps/H

z]

FDMA−bound (s=1)

FDMA−bound (s=2)

FDMA−bound (s=3)

FDMA−bound (s=4)

FDMA−bound (prorated)

OFDM BC DLC region

Figure 4.6: OFDM BC DLC region for 16 subcarriers at 10 dB. Thelower bound is shown fordifferent values of the parametersand the prorated FDMA strategy

1 and term 2. The optimum choice ofs is not obvious and depends on the SNR regime as well

as the number of subcarriers. The bounds from Lemma4.10are illustrated in Figs.4.5and4.6

for different values ofsand i.i.d. Rayleigh fading as specified in Corollary4.11. In Fig. 4.5the

low SNR case is depicted. It can be seen that the lower bound achieves nearly the entire region.

This is due to the fact that in the low SNR regime the optimum transmission strategy for each

user consists of serving only the best subcarrier. Regardless of the subcarrier assignment among

users this coincides with the strategy used for the first termin (4.38) even if not perfect but only

ordinal information is present. The remaining gap stems from the second term and the fact that

users can ”collide”, i.e., have a common best subcarrier. Incontrast, in Fig.4.6 the high SNR

scenario is illustrated. The bound improves ass increases up tos= 4. This is obviously due to

the fact that it is not optimal to support the entire rate onlyon one subcarrier, even if a user has

only one best subcarrier. Froms = 5 on the bound degrades once again since the second term

becomes predominant. Note that for the case that both users have similar rate requirements,

i.e. thesum DLCcase, the bound achieves a major part of the DLC. The remaining gap on

the axes is much bigger. However, the gap on the axes can be reduced using only the second

term (thus making the conditioning of the pdf needless) and by partitioning the subcarriers not

equally among users. Then the second sum of the second term in(4.38) does not have⌊K/M⌋addends butKm addends for each user with

∑Mm=1 Km = K. So it is especially reasonable to share

the subcarriers proportional to the users rate requirements such thatKm = Rm/∑M

m=1 Rm. This

is illustrated with the curve calledprorated FDMA. The discontinuity stems from the switching

of the subcarrier allocation since this is a discrete problem.

The FDMA-bounds with the used concept of ordinal channel state information seem to have

positive implications on the necessary feedback. Unfortunately they are not constructive since


it is not possible to allocate a fixed rate budget in each fading state based on the ordinal channel

state information. In order to do so, the exact channel gain realization has to be known. If the

fixed rate budget is transformed to a fixed power budget the achieved rate becomes a random

variable once again albeit having a smaller variance.

While the used FDMA strategies are suboptimal in general we characterize in the following

under which conditions FDMA is indeed the optimum transmit strategy.

4.3.2 Conditions for Optimality of FDMA Transmission

In the previous section lower bounds on the OFDM BC DLC regionbased on FDMA were

developed. The gap compared to the real OFDM DLC region stemsmainly from the fact that

a rate allocation independent of the current channel realization is used and the subcarrier as-

signment is very simple. But even without these simplifications FDMA transmission itself is

suboptimal in general. In this section we re-consider the sum power minimization problem for

a single fading state and derive conditions for the optimality of a specific FDMA subcarrier as-

signment. In order to stay as general as possible we state theresults in the more general OFDM

MAC setting implying individual user-specific powers. Thiscovers the OFDM BC scenario as

a special case by choosing equal weights for all users, i.e.λ1 = ... = λM. Then the weighted

sum power minimization problem stated in Section3.3.1reads as:

minλTp subj. to R ∈ C(g, p1, ..., pM). (4.53)

Hereλ ∈ RM+ is an arbitrary vector of power weights andp ∈ RM

+ is the vector of the users’

transmit power budgets. Given a set of rate requirementsR1, ..., RM the following Lemma gives

necessary and sufficient conditions for the optimality of FDMA transmission.

Lemma 4.12. The solution to(4.53) with given positive rate requirementsR ∈ RM+ has an

FDMA structure with user mk being served exclusively on subcarrier k if and only if thereexists

a rate allocation with∑K

k=1 Rm,k = Rm for all m and a set of Lagrangian multipliersµ ∈ RM+ such

that

mink

λmk

µmkgmk,kexp

(

Rmk,k) ≤ min

m,k

1µm

([

λm

gm,k− λmk

gmk,k

]+

+min

[

λm

gm,k,λmk

gmk,k

]

exp(

Rmk,k)

)

.

Proof. Assume that it is optimal to serve usermk exclusively on subcarrier k while all other

usersm, mk miss out on this subcarrier.

Then the first KKT condition from (3.30) for usermk yields

µmk + ψmk,k =

M∑

s=m

cs,k exp(

Rmk,k)

=λmk

gmk,kexp

(

Rmk,k)


and thus

1+ψmk,k

µmk

=λmk

µmkgmk,kexp

(

Rmk,k)

. (4.54)

For users m withπ−1k (m) > π−1

k (mk) we get equivalently

1+ψm,k

µm=

λm

µmgm,kexp

(

Rmk,k)

(4.55)

and for m withπ−1k (m) < π−1

k (mk) we have

1+ψm,k

µm=

λm

µmgm,k− λmk

µmgmk,k+

λmk

µmgmk,kexp

(

Rmk,k)

, (4.56)

whereπ−1k (·) is the inverse mapping ofπk(·). If the rate requirements are positive for at least one

user, i.e.Rm > 0 for at least onem, then it follows thatψmk,k = 0 for at least onek. Thus (4.54)

yields

1 = mink

λmk

µmkgmk,kexp

(

Rmk,k)

. (4.57)

Sinceψmk,k ≥ 0 for all k andm, (4.57) together with (4.55) and (4.56) yields

mink

λmk

µmkgmk,kexp

(

Rmk,k) ≤ min

m,k

1µm

([

λm

gm,k− λmk

gmk,k

]+

+min

[

λm

gm,k,λmk

gmk,k

]

exp(

Rmk,k)

)

.

which states the necessary part. Sufficiency follows immediately from the fact that non-negative

Lagrangian multipliers can be found so that the KKT conditions are fulfilled. This yields the

desired result. �

Corollary 4.13 (Necessary Condition). A necessary condition for a specific subcarrier assign-

ment to be FDMA-optimal is that each subcarrier k receiving anon-negative amount of power

is assigned to a user mk such that

λmk

µmkgmk,k≤ min

m

λm

µmgm,k. (4.58)

Proof. This follows immediately from the fact that for all subcarriers k receiving power the

KKT conditions yield

λmk

µmkgmk,kexp

(

Rmk,k)

= 1 ≤ 1+ψm,k

µm=

λm

µmgm,kexp

(

Rmk,k)

because of the complementary slackness conditions. �

Lemma4.12and Corollary4.13involve not only the rate allocations but also the Lagrangian

multipliers µ1, ..., µM. Thus the conditions seem complicated to check. However, once an

FDMA subcarrier assignment is chosen the optimal rate allocation per user follows from rate


water-filling as specified in (4.2) and (4.3) and the Lagrangian multipliersµ1, ..., µM are de-

termined by the corresponding water-filling levels. For theconsidered broadcast scenario the

necessary condition in (4.58) simplifies to

µmkgmk,k ≥ maxm

µmgm,k. (4.59)

so that the user with the maximum weighted channel gain has tobe chosen. It remains to be

mentioned that the conditions given in Lemma4.12can be equivalently derived for weighted

sum rate-maximization stated as Problem1 in Section3.1 where the weights are known in

advance. In this case the optimum power allocation consistsof a water-filling procedure with

weighted noise terms and a weighted water-filling level [78],[10].

4.3.3 Sum Power Minimization under FDMA Constraints

Lemma4.12gives conditions for the optimality of FDMA transmission. However, except for

the low SNR regime it is very unlikely that FDMA is indeed optimal. Intuitively the probability

for this event decreases as the frequency-selectivity combined with the number of subcarriers,

the number of involved users or the rate requirement and thusthe transmit SNR per user in-

creases. For this reason we consider the sum power minimization problem under individual rate

constraints with exclusive subcarrier assignment in the following. Introducing an additional

FDMA constraint in Problem2, we arrive at

minK∑

k=1

Pk

subj. toK∑

k=1

νm,k log(

1+ Pkgm,k) ≥ Rm ∀ m

M∑

m=1

νm,k ≤ 1 νm,k ∈ {0, 1} ∀ m, k

(4.60)

whereνm,k are binary variables reflecting the exclusive subcarrier allocation. Unfortunately,

(4.60) is a non-convex problem since due to the FDMA constraints (νm,k ∈ {0, 1}) an optimiza-

tion over a discrete and thus non-convex set is involved. Thecombinatorial structure makes

the problem complex. A brute-force solution consists of rate water-filling for allMK possible

subcarrier assignments which is obviously prohibitive. Thus a common possibility to deal with

discrete constraints is to use a convex relaxation approachrendering the problems at hand con-

vex at the cost of increasing the feasible set [78, 79, 80]. For example, Hoo et al. proposed

to apply this approach to weighted sum rate-maximization for OFDM BCs in [78]. Neverthe-

less, to solve the relaxed problem demanding convex optimization tools have to be used and

the solution has to be projected onto the feasible set of the original non-convex problem. In

contrast to the continuous relaxation technique we will focus on the methodology developed


in Section3.2.3of the previous chapter. In fact the presented approach can be interpreted as

an optimization in the dual domain. Dual methods are well known in optimization theory and

were proposed recently for the optimization of problems under non-convex FDMA constraints

in [81]. The authors showed that the duality gap becomes zero if a certain property termed

time-sharing propertyis fulfilled. This criterion was made more precise in [82]. It was argued

that for large numbers of subcarriers the time-sharing property is approximately fulfilled and

thus dual methods yield excellent results. Independently from this work Seong et al. presented

a dual approach for the problem at hand in [83]. In contrast, the subsequent framework has a

more axiomatic nature and bases on three crucial propertiesof the rate functions. It is more

general and can be applied also to scenarios with discrete power allocations arising e.g. in the

context of bit loading.

In order to apply the ideas from Section3.2.3we introduce the set

GFDMA(g)def=

{

(R,P) ∈ RM+1+ : R ∈ CFDMA(g,P)

}

, (4.61)

where the set of all rate tuples achievable with sum powerP under an FDMA constraint is given

by

CFDMA(g,P)def=

⋃

M∑

m=1νm,k≤1 k∈K ,νm,k∈{0,1}

⋃

∑

m,kpm,k≤P

R ∈ RM

+ : Rm ≤1K

K∑

k=1

νm,k log(1+ gm,kpm,k)

.

Thus (4.61) is the analogon toG(g) defined in (3.15) and contains all rate-power tuples which

are achievable using an FDMA strategy. Unlike (3.15) the setGFDMA(g) is not convex in general.

Nevertheless we consider in analogy to (3.16) the following problem:

maxM∑

m=1

µmRm − λP subj. to (R,P) ∈ GFDMA(g) (4.62)

In contrast to sum power minimization involving superposition coding as studied in Section

3.2.3it is not sufficient to find appropriate non-negative Lagrangian multipliersµ1, ..., µM for

(4.62) such thatRm ≥ Rm for all m in order to solve (4.60). Such a solution characterizes a

stationary point but due to the non-convexity global optimality can not be guaranteed in this

case.

In order to proceed further we define the solution to (4.62) for a set of parametersµ andλ

as

(R1(µ, λ), ...,RM(µ, λ),P(µ, λ))def= arg max

GFDMA(g)

M∑

m=1

µmRm− λP. (4.63)

We need some important properties summarized in the subsequent lemma:

Lemma 4.14.Consider the optimization problem in(4.63). Then(R1(µ, λ), ...,RM(µ, λ),P(µ, λ))


has the following three properties:

1. (normalization): Rm(0, λ) = 0 for all m ∈ M and anyλ > 0

2. (monotonicity): Let µ(1), µ(2) be two vectors withµ(1)m > µ

(2)m for some m andµ(1)

n = µ(2)n for

n , m. Then

(a) Rm(µ(1), λ) ≥ Rm(µ(2), λ) and Rn(µ(1), λ) ≤ Rn(µ(2), λ) for n , m

(b) P(µ(1), λ) ≥ P(µ(2), λ)

3. (unboundedness): Let µ = [µ1, ..., µM]T . Then

(a) limµm→∞Rm(µ, λ) = ∞ for user m and

(b) limµm→∞Rn(µ, λ) = 0 for users n, m

Proof. In order to prove the properties recall (4.62). Obviously (4.62) decouples toK indepen-

dent problems since

maxM∑

m=1

µmRm − λP subj. to (R,P) ∈ GFDMA(g)

=1K

K∑

k=1

(

maxm

maxpk

µm log(1+ gm,kpk) − λpk

)

=1K

K∑

k=1

maxm

fm,k(µm, λ), (4.64)

where

fm,k(µm, λ) = maxpµm log(1+ gm,kp) − λp (4.65)

denotes the objective per user on subcarrierk. The valuefm,k(µm, λ) is easy to obtain since it is

the maximum over a concave function. Note that the FDMA constraint is implicit in (4.64) by

the maximum operation ofm.

1. (normalization) We havefm,k(0, λ) = 0 for all m andk with

arg maxpµm log(1+ gm,kp) − λp = 0,

i.e. no power is spent. This yields normalization.

2. (monotonicity) Property (b) follows from the fact that

arg maxm

µ(1)m log(1+ hm,kp) − λp ≥ arg max

mµ(2)

m log(1+ hm,kp) − λp.

Property (a): Sincefn,k(µn, λ) is independent ofµm for n , m all other powers remain

unchanged. Further from

fm,k(µ(1)m , λ) ≥ fm,k(µ

(2)m , λ)


Algorithm 7 OFDMA Power Minimization Algorithm

(1) initialize µ(0) = 0, λ arbitrary (w.l.g.λ = 1)while desired accuracy not reacheddo

for m=1 to M do(2) find minimumµ

(n+1)m such thatRm(µ, λ) ≥ Rm evaluating (4.63) and using bisection

end forend while

it follows that

Rm,k(µ(1), λ) ≥ Rm,k(µ

(2), λ)

and

Rn,k(µ(1), λ) ≤ Rn,k(µ

(2), λ)

for all n , m. Since this holds for allk the claim follows.

3. (unboundedness) This follows since

limµm→∞

arg maxm

µm log(1+ hm,kp) − λp = ∞

and

limµm→∞

fm,k(µm, λ) = ∞ > fn,k(µn, λ)

for any finiteµn andn , m. Thus userm gets infinite power on eachk while all n , m

miss out which is the desired result.

�

In Section3.2.3, where the optimum resource allocations without FDMA constraints are

studied, Alg. 2 exploits exactly the three key properties normalization, monotonicity and un-

boundedness in order to find a stationary point. Since these properties also hold for the FDMA

case an algorithm similar to Alg.2 can be developed. A cyclic adaptive increase of the factors

µm leads to a stationary power allocation achieving the desired set of ratesR. As mentioned sta-

tionarity is not equivalent to global optimality since the problem is not convex. The procedure

is summarized in Alg.7. Further the following Lemma ensures convergence.

Lemma 4.15. Alg. 7 generates a monotonously increasing sequence of multipliers µ(n) and

powers P(µ(n), 1) converging to a fixed pointµ∗ such that the rate requirements are met for all

users, i.e. Rm(µ∗, 1) ≥ Rm for all m.

Proof. Since the system is not interference limited and due to Property 3 of Lemma4.14there

always exists aµ∗ such thatRm(µ∗, 1) ≥ Rm for all m. Now define the mappingT : RM+ 7→ RM

+

representing one cyclic update in Alg.7 so thatµ(n+1) = T(

µ(n))

. The mappingT is order

preserving, i.e.

T(

µ(1))

≤ T(

µ(2))

for µ(1) ≤ µ(2).


With the initializationµ(0) = 0M < µ∗ this yields

Tn(

µ(0))

≤ Tn (µ∗) = µ∗

bounding the sequence from above and thus we have

limn→∞

Tn(

µ(0))

= µ∗.

The monotonicity of the sequence of powersP(

µ(n), 1)

follows immediately from Property 2 of

Lemma4.14which is the desired result. �

Since (4.63) is nothing but the evaluation of the Lagrangian dual function, Alg. 7 can be

interpreted as a dual optimization method where the modification of the Lagrangian multipliers

is carried out in a cyclic manner. Note that Alg.7 needn’t necessarily maximize the dual

function since the stopping criterion is the fulfillment of the rate constraints. This is in contrast

to [83], where the ellipsoid method was used to maximize the dual. Another difference is that

based on the three properties in Lemma4.14it is possible to generalize Alg.7 to the case of

discretized power allocations.

Extension to Discrete Power Allocations

Note that apart from the properties of Lemma4.14Alg. 7 exploits the fact that for fixed La-

grangian multipliersµ andλ the dual in (4.63) can be easily evaluated using (4.65) due to the

concavity of its argument. For this reason the procedure canbe applied in all scenarios as long

as the rate function is concave. It even worksif only certain points on a concave rate function

can be attained. This is the case for all settings with monotonously decreasing marginal utility.

Then in each step the solution has to be projected to the next feasible power allocation. Note

that further the rate function has to be unbounded (property3) which calls for infinitely many

possible discrete allocations. This means that Alg.7 can be also applied for bit loading sce-

narios opening interesting possibilities for real world applications. One way to model such a

discrete allocation according to the mentioned constraints is to define the feasible set of powers

for usermon subcarrierk as

Pm,kdef=

{

p ∈ R+ : p =1

gm,k

(

2Rm,k − 1)

, Rm,k ∈ N}

, (4.66)

which corresponds to the case of integer bit allocations.

An Approach Based on the Necessary FDMA Optimality Condition

For the sake of completeness a heuristic approach proposed in [10] has to be mentioned. The

idea here is to exploit the necessary FDMA optimality conditions of Lemma4.12specified in


Algorithm 8 OFDMA based on FDMA optimality conditions(1) use Alg. 4 to calculate optimal (incl. superposition coding) resource allocation andLagrangian multipliersµ1, ..., µM.(2) assign subcarrierk to usermk according to

mk = arg maxm

µmgm,k (4.67)

for all k(3) perform rate water-filling such thatR is met for allm

(4.59) to choose a subcarrier assignment. Subsequent to the assignment rate water-filling can

be performed. Note that the Lagrangian multipliersµ1, ..., µM occur in (4.59) being unknown

at the time of decision. In order to calculate the multipliers, Alg. 4 performing iterative rate

water-filling can be used. This procedure is summarized in Alg. 8.

Illustration of Numerical Examples

Figure4.7contains a comparison of all algorithms comparing the optimum achieved by Alg.4

with the FDMA solution of Alg.7 for the continuous case and the discrete case representing bit

loading. For the discrete case integer bit allocations as defined in (4.66) were used. The upper

left figure depicts the case withL = K = 16 subcarriers, the upper right figure the case with

L = K = 32 subcarriers and the figures at the bottom illustrate the case withL = K = 128.

Except forK = 16 the figures contain additionally the curve achieved by Alg. 8 based on

the FDMA optimality conditions. For all settingsM = 4 users with equal rate requirements

(absolute fairness) indicated by the value on the ordinate had to be served and 1000 Monte

Carlo runs were simulated.

It can be observed that in all cases the FDMA solution performs very close to the optimal.

The remaining gap decreases with an increase ofK. This reflects the argument in [81]. For

K = 128 even in the enlargement both curves are hardly distinguishable. The discrete version

of Alg. 7 according to (4.66) produces a non-smooth curve for small values ofK. This is due

to the fact that the minimum rate to be allocated is 1/K bit/s/Hz because of the discretization.

The curves become smoother asK increases but a constant offset due to the finite granularity

remains. However, forK = 128 the performance offset becomes almost negligible at high SNR.

The plot for K = 16 does not contain a curve for Alg.8. This is due to the fact that

the subcarrier assignment according to (4.67) might not necessarily yield a feasible solution.

Especially for smallK there is a non-negligible probability that a user does not receive any

subcarrier rendering the problem infeasible. For largerK the probability of this event decreases.

However the performance degrades as the rate requirements increase forK = 32 andK = 128

because too view subcarriers might be assigned to some of theusers which causes a high power

penalty at high SNR due to the concavity of the rate function.


−10 −5 0 5 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

transmit SNR [dB]

Rat

e/us

er [b

ps/H

z]

optimum (Alg. 4)

OFDMA (Alg. 7)

OFDMA discrete (Alg. 7)

−10 −5 0 5 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

transmit SNR [dB]

Rat

e/us

er [b

ps/H

z]

optimum (Alg. 4)OFDMA (Alg. 7)OFDMA discrete (Alg. 7)OFDMA based on Alg. 8

−10 −5 0 5 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

transmit SNR [dB]

Rat

e/us

er [b

ps/H

z]


enlargement

5 6 7 8 9 100.75

0.8

0.85

0.9

0.95

1

transmit SNR [dB]

Rat

e/us

er [b

ps/H

z]


Figure 4.7: Comparison of FDMA algorithm (Alg. 7) with continuous and discrete powerallocation with optimum solution (Alg.4) forL = K, K=16 (top left), K=32 (top right) andK=128 (bottom). The right figure at the bottom contains an enlargement of the area marked inthe figure on the left


4.3.4 Performance Bounds Via Duality Gap

As the numerical results in the last section indicate OFDMA schemes can perform very well

implying only a minimal performance loss compared to the optimum. In this section we provide

a bound on the performance loss for all rate (and thus power) allocations meeting the users’ rate

constraintsR for an arbitrary fixed fading realizationg. All variables refer to the OFDM MAC.

To make it more precise consider an arbitrary feasible rate allocation R′1,1, ...,R′M,K so that

∑Kk=1 R′m,k = Rm for all m. As a result of Chapter3 this rate allocation implies a unique power

allocationp′1,1, ..., p′M,K using the bijective mappings

pπk(m),k =1

gπk(m),k

(

exp(

Rπk(m),k) − 1

)

exp

∑

n<m

Rπk(n),k

., (4.68)

where

πk(·) :1

gπk(1),k≥ ... ≥ 1

gπk(M),k. (4.69)

Further letR∗1,1, ...,R∗M,K with p∗1,1, ..., p

∗M,K and sum powerP∗ =

∑Mm=1

∑Kk=1 p∗m,k be the opti-

mum power allocation and thus the solution to the sum power minimization problem with rate

constraintsR

min P subj. to R ∈ C(g,P). (4.70)

We would like to knowhow goodthe allocationR′ = [R′1,1, ...,R′M,K]T is compared to the opti-

mum, i.e. how much power is spent exceeding the necessary power P∗ in order to achieve the

set of target ratesR. The following theorem yields a bound on the power loss.

Theorem 4.16.LetR′ = [R′1,1, ...,R′M,K]T be an arbitrary feasible rate allocation with

∑Kk=1 R′m,k =

Rm for all m = 1, ...,M. Further let P(R′) =∑M

m=1

∑Kk=1 pm,k(R′) be the corresponding sum

power with pm,k(R′) given in(4.68) and let P∗ be the solution to(4.70).

Then the power gapΓ(R′) = P(R′) − P∗ is upper bounded by

Γ(R′) ≤M∑

m=1

K∑

k=1

R′m,k

(

Θm,k −minlΘm,l

)

(4.71)

where

Θm,k =

M∑

s=uu:πk(u)=m

cs,k exp

∑

n≤s

R′πk(n),k

with

cm,k =

1gπk(m),k

− 1gπk(m+1),k

m= 1, ...,M − 1

1gπk(M),k

m= M

andπk(·) is given in(4.69).

Proof. As shown in Section3.3.1the problem in (4.70) is convex. The Lagrangian dualg(µ,ψ)


yields a lower bound on the optimum for every feasible rate allocation. Hence we have

Γ(R′) = P(R′) − P∗

≤ P(R′) − g(µ,ψ)

= P(R′) − infRL(R, µ,ψ), (4.72)

where the Lagrangian of (4.70) is given by

L(R, µ,ψ) =M∑

m=1

K∑

k=1

pm,k(R) +M∑

m=1

µm

Rm−K∑

k=1

Rm,k

−M∑

m=1

K∑

k=1

ψm,kRm,k. (4.73)

The idea is to chooseµ ∈ RM+ andψ ∈ RMK

+ such thatR′ is a solution to arg infRL(R, µ,ψ). As

a consequence the gap can be expressed in terms of the considered allocation. UsingR′ in the

stationarity condition of (4.73) yields

M∑

s=uu:πk(u)=m

cs,k exp

∑

n≤s

R′πk(n),k

= µm + ψm,k. (4.74)

Thus we arrive at

Γ(R′) ≤ P(R′) − L(R′, µ,ψ)

≤ P(R′) −M∑

m=1

K∑

k=1

pm,k(R′)

︸︷︷︸

0

−M∑

m=1

µm

Rm −K∑

k=1

R′m,k

︸︷︷︸

0

+

M∑

m=1

K∑

k=1

ψm,kR′m,k

=

M∑

m=1

K∑

k=1

ψm,kR′m,k

=

M∑

m=1

K∑

k=1

R′m,k

M∑

s=uu:πk(u)=m

cs,k exp

∑

n≤s

R′πk(n),k

− µm

, (4.75)

whereµ ∈ RM+ andψ ∈ RMK

+ . The last equality follows from (4.74). Obviouslyµm has to be

chosen as large as possible for allm in order to tighten the bound. By inspection of (4.74) it

becomes clear that

µm ≤ minl

M∑

s=uu:πl (u)=m

cs,l exp

∑

n≤s

R′πl (n),l

,

since any other choice results inψm,k < 0 for at least onem andk. This yields the desired

result. �

Corollary 4.17. For any FDMA allocation with R′mk,k≥ 0 and R′m,k = 0 for m , mk and all k

4.4. Summary 83

Theorem4.16simplifies to

Γ(R′) ≤K∑

k=1

R′mk,k

(

Θmk,k −minlΘmk,l

)

where

Θm,k =

1gmk,k

exp(

R′mk,k

)

+

(

1gm,k− 1

gmk,k

)

if u > r : mk = πk(u), m= πk(r)

1gm,k

exp(

R′mk,k

)

if u ≤ r : mk = πk(u), m= πk(r)

andπk(·) is given in(4.69).

The expressions are very simple to evaluate. Note that the rate water-filling condition is

reflected in (4.71): The gap becomes zero if and only if the rate allocations arewater-filling

solutions with respect to each other. In this case the expressionΘm,k −minl Θm,l becomes zero

for all m, k which receive power. All others do not play a role in the expression. In other words,

the water-filling level must be equal for all subcarriers receiving power. Using the duality gap in

order to bound the performance has been proposed recently byYu and Cioffi in [84], where they

derived performance bounds for single user constant power water-filling. The crucial point is

that there must exist a set of non-negative Lagrangian multipliers so that the arbitrary resource

allocation is a solution for the evaluation of the dual. For convex problems this is always the

case. The situation changes if non-convex problems are considered [11] where not necessarily

a set of dual parameters exists for all possible resource allocations.

4.4 Summary

Throughout this chapter we studied different aspects of delay-limited transmission over OFDM

broadcast channels. First we analyzed the behavior at low and high SNR and derived so the

minimum bit-energy at which delay-limited transmission ispossible. It turns out that the delay

spread, i.e. the number of paths, is the predominant factor of influence on the OFDM DLC

at low SNR while at high SNR the fading distribution becomes more important. Then we

derived conditions on the joint fading distribution for theexistence of a non-zero single user

delay-limited capacity characterizing the fading distributions commonly termedregular. We

subsequently proposed an algorithm to evaluate the OFDM BC DLC region. To this end we re-

ferred to the power minimization algorithm presented previously in Chapter3. The evaluation,

however, is prohibitive for two reasons. First, the expression for the DLC region is only implicit.

Second, the distribution of the power necessary to achieve aset of rates can not be expressed

analytically so that it has to be approximated by Monte-Carlo runs. In the latter part of our

analysis we focused on FDMA transmission schemes. Lower bounds on the OFDM BC DLC

region, which exploit only ordinal channel state information, were derived. These bounds are

not very practical. We introduced an iterative algorithm tofind the minimum power supporting


a given set of rates under FDMA constraints for a fixed fading realization. It can be interpreted

as a dual optimization method adaptively modifying the Lagrangian multipliers. Further, it can

be easily generalized to the case of discrete rate allocations reflecting the situation occurring

for bit-loading. Both strategies perform very well and do not show a surprisingly small perfor-

mance loss compared to the optimal strategy. Finally, we derived bounds based on a duality gap

expression in order to characterize the performance loss ofan arbitrary resource allocation.

Chapter 5

Resource Allocation for Multiple-Antenna

OFDM Systems with Time-Invariant

Channels

Systems with multiple antennas at both sides of the link havebeen known to achieve high gains

in terms of spectral efficiency since the pioneering work of Telatar [25] and Foschini and Gans

[26]. However, while MIMO point-to-point communication is quite well understood, the uplink

and downlink of such systems is significantly more complicated. The spatial degrees of freedom

make an analysis very complex and cause the elementary frequency flat MIMO BC to not be

degraded any more. For this reason it was not until very recently that its capacity region was

rigorously established [36]. In contrast to the single antenna case, a convex formulation in the

broadcast scenario of most resource allocation problems isnot possible. As a consequence one

relies on the MIMO duality relations presented in Section2.3.3and the convex re-formulations

in the dual MAC setting.

In this chapter we extend our study of single antenna OFDM multi-user systems with time-

invariant channels to multiple antennas. We mainly focus onthe broadcast scenario implying

a sum power constraint. However, since in all cases the analysis and the corresponding algo-

rithms take place in the dual MAC scenario the results hold for the uplink as well. The three

reference problems introduced in Section3.1 in the context of single antenna systems serve as

a guideline. Clearly the motivation for and importance of the reference problems carries over

to multi-antenna scenarios. We begin with an analysis of theweighted rate sum-maximization

problem in Section5.1 where we additionally derive conditions for the optimalityof FDMA

transmission. Narrowing the view to sum capacity the influence of the number of users on the

expected throughput is investigated for the MIMO-OFDM downlink with and without FDMA

constraints, if a sum power constraint has to be met in each block fading state. Section5.2

contains an analysis of Problem2. For the important special case of a sum rate requirement we

then develop an algorithm based on iterative rate water-filling. Finally we turn towards Prob-

85

86 Chapter 5. Resource Allocation for Multiple-Antenna OFDM Systems

lem 3 including minimum rate requirements in Section5.3 and additionally comment on the

incorporation of FDMA constraints in Section5.4. Parts of this chapter have been published

previously in [12, 13, 14, 15, 16].

5.1 Weighted Rate-Sum and Throughput Maximization

In this section we first consider Problem1. In contrast to the single antenna case in Chapter

3 where a MAC approach as well as a BC approach was presented no way is known to state

the problem in the broadcast setting in convex form. Thus theanalysis is carried out in the

MIMO-OFDM MAC. The resulting optimum transmit policies canbe easily transformed to the

broadcast scenario using the duality relations (2.40) introduced in Chapter2. After some com-

ments on algorithmic aspects we subsequently derive optimality conditions for FDMA which

have a similar structure to the conditions in Section4.3.2for single antenna scenarios. We then

focus on the sum capacity case and derive the scaling behavior as the number of active users in

the system increases.

5.1.1 Weighted Rate-Sum Maximization

Recall Problem1. It is given by

max µTR subj. to R ∈ C(H, P) (5.1)

whereµM+ is a vector of user-specific weights andP denotes the available sum power budget.

For notational convenience we writeHk = [HT1,k, ...,H

TM,,k]

T for the matrix of stacked channels

on subcarrierk andH = [HT1 , ...,H

TK]T for the stacked matrix of all channels, respectively. The

formulation takes place in the uplink. As described, the duality relations in (2.40) can be used

to transform the optimum transmit strategy from the dual MACto the broadcast scenario. We

omit the subscript·MAC used throughout Chapter2 in the following. Using (2.35) and (2.38)

from Chapter2 the capacity region can be rewritten as

C(H, P) =⋃

∑Mm=1 Pm≤P

⋃

Qm,k�0∑K

k=1 tr(Qm,k)≤Pm ∀∈M

K∑

k=1

R(Hk,Q1,k, ...,QM,k) (5.2)

with R(Hk,Q1,k, ...,QM,k) given in (2.33) and the addition of sets defined in Definition2.2. For

each elementary transmit strategy, i.e. a fixed set of transmit covariance matricesQ1,k, ...,QM,k

the achievable rate regionR(Hk,Q1,k, ...,QM,k) is a polymatroid. As a consequence the achiev-

able rate region∑K

k=1R(Hk,Q1,k, ...,QM,k) consisting of the sum of polymatroids is a polyma-

troid. It remains to point out that in contrast to the single antenna case it does not suffice to fix

a power allocation per user and subcarrier in order to define an elementary transmission policy.

5.1. Weighted Rate-Sum and Throughput Maximization 87

Rather thespatialsecond order statistics of the input signal are necessary tospecify it uniquely.

Using TheoremA.8 from the Appendix the optimum vertex and thus the optimum decoding

order in (5.1) is given by

π(·) : µπ(1) ≥ µπ(2) ≥ ... ≥ µπ(M) (5.3)

independent of the transmit strategy: It is optimal to decode the user with lowest weightπ(M)

first, then userπ(M − 1) and so on till the userπ(1) with highest priority - not suffering inter-

ference any more - is decoded finally. The encoding order in the BC is reversed. Note that

in contrast to the single antenna case an optimum decoding order can not be derived from the

channel realizations.

As a consequence the optimization problem results in an optimization over corner points of

an uncountable set of polymatroids where the decoding orderis determined by the polymatroid

structure of the partial regions. WithAmk

def= I +

∑mn=1 HH

π(n),kQπ(n),kHπ(n),k the individual rates can

be expressed as

Rπ(m) =

K∑

k=1

log∣∣∣Am

k

∣∣∣ − log

∣∣∣Am−1

k

∣∣∣ ∀ m ∈ M.

With coefficients

cm =

−µπ(1) m= 0

µπ(m) − µπ(m+1) m= 1, ...,M − 1

µπ(M) m= M

we can rewrite the optimization problem in (5.1) as

maxC(H,P)

M∑

m=1

µmRm = maxC(H,P)

M∑

m=0

cm

K∑

k=1

log∣∣∣Am

k

∣∣∣ . (5.4)

Since each addend in (5.4) is jointly convex inQm,k for all m and the objective function is a

linear combination of convex functions with nonnegative weight factorscm, (5.4) is a convex

optimization problem. Note that the convexity is guaranteed by the optimal decoding order,

resulting from the polymatroid structure of the partial capacity regions. The appropriate recom-

bination of terms yields the non-negative sum in (5.4). If all weightsµ1, ..., µM are distinct, any

other decoding order leads to at least one negative addend and thus a non-convex objective.

The Lagrangian of (5.4) is given by

L({

Qm,k}M,Km,k=1 ,

{

Φm,k}M,Km,k=1 , λ

)

=

M∑

m=0

cm

K∑

k=1

log∣∣∣Am

k

∣∣∣+λ

P−M∑

m=1

K∑

k=1

tr(Qm,k)

+

M∑

m=1

K∑

k=1

Φm,kQm,k

whereΦm,k � 0andλ ∈ R+ are Lagrangian multipliers corresponding to the positive-semidefiniteness

of Qm,k and the sum power constraintP, respectively.

Due to convexity the corresponding Karush-Kuhn-Tucker (KKT) conditions are necessary


and sufficient for optimality. These are given by the following set ofequations.

M∑

s=m

csHπ(m),k(Ask)−1HH

π(m),k − λI +Φm,k = 0 ∀k (5.5)

P−M∑

m=1

K∑

k=1

tr(Qm,k) ≥ 0

λ

P−M∑

m=1

K∑

k=1

tr(Qm,k)

= 0

Qm,k � 0 ; Φm,k � 0 ; λ ≥ 0 ; tr(Qm,kΦm,k) = 0 ∀m, k (5.6)

The polymatroid structure of the MIMO MAC implying these results has been known quite

some time [85, 86]. It naturally carries over to the MIMO-OFDM case since the sum of polyma-

troids is a polymatroid or equivalently since a MIMO-OFDM MAC is a blockdiagonal MIMO

MAC.

5.1.2 Algorithmic Approaches

The optimization problem in (5.4) can be solved by standard convex optimization methods in

general. However, the number of involved optimization variables can be quite large. This is the

case especially for OFDM, where the driving factor is the number of subcarriers which easily

exceeds a thousand in modern systems. For this reason a more efficient algorithmic solution

is of fundamental interest. The area of algorithm design forweighted rate sum-maximization

as well as throughput maximization has experienced a considerable boost very recently. We

review briefly the existent approaches.

The first algorithm for weighted rate sum-maximization was proposed in [87] where the

authors applied the steepest descent principle taking in each step a convex combination of the

former covariance matrices and the eigenvector corresponding to the maximum eigenvalue of

the gradient with respect to one user. The sum power constraint is kept implicit this way.

However, the convergence speed is quite poor. The main disadvantage in the context of MIMO-

OFDM systems is that an eigenvalue decomposition of each user’s block-diagonal gradient

matrix has to be performed in each step. As the number of subcarriersK can be quite large, this

is computationally prohibitive even for a moderate number of subcarriers. About the same time

an iterative water-filling algorithm for MIMO MAC sum rate maximization was proposed in

[86] which was non-trivially extended to a sum power constraintcovering the broadcast scenario

in [88]. The main difference between the sum rate case and arbitrary weights is that the first

KKT condition in (5.5) consists of a single addend. This fact can be exploited and leads to the

concept ofeffective channelsmaking an iterative cyclic update of all users’ transmit covariance

matrices possible. The convergence speed is much better andnot so much influenced by the

number of involved users and subcarriers. However, this comes at the cost of an increased


need for memory as forM usersM − 1 old covariance matrices have to be stored [88]. The

iterative water-filling approach was further modified in [89, 90]. In [89] the authors proposed

to update only two users at the same time speeding up convergence and in [90] it was suggested

to use a parabola-approximation for the sum rate in order to maximize the update step-size.

These approaches overcome the increasing need for memory. For the special case of MISO

systems a modified iterative water-filling algorithm solving the weighted rate sum problem was

proposed in [91] for flat channels and then generalized to OFDM using dual decomposition in

[92]. Unfortunately the approach can not be generalized to multiple antennas. The idea to use

dual decomposition for sum rate maximization was in turn introduced in [93].

In a very recent second line of work it was proposed to use (conjugate) projected gradient

algorithms for all problems. The idea occurred first for sum rate maximization in [94] was

generalized to the general case with arbitrary weights in [95]. This idea was further developed

in [90, 96, 97] leading to the best known weighted rate sum maximization algorithms. The

connection to mean square error-minimization and linear precoding was pointed out in [98,

99, 96]. The authors proposed to optimize the spatial transmit filters instead of the covariance

matrices opening the possibility to treat linear precodingas well as non-linear interference pre-

cancellation. However, for linear precoding it was not possible to prove convergence to the

global optimum for these cases due to the non-convex structure of the problem up to now.

5.1.3 Optimality Conditions for FDMA Transmission

In order to simplify signal processing we characterize cases in which the exclusive use of sub-

carriers is optimal, i.e. tr(Qmk,k) ≥ 0 and tr(Qm,k) = 0 for all k,m , mk. This avoids the use

of complex successive pre- and decoding operations. The conditions are generalizations of the

conditions in Lemma4.12stated for the single antenna case in Section3.3.1. The following

theorem states necessary and sufficient conditions for the optimality of pure FDMA for the

MIMO-OFDM case. The derivation starts with Problem1 instead of Problem2.

Theorem 5.1.Assume that the rate rewards are ordered, i.e.µ1 ≥ ... ≥ µM. Let the index of the

user transmitting exclusively on subcarrier k be mk. Exclusive transmission on each subcarrier

(FDMA), i.e. allocating the whole powerP to the set m1, ...,mK, is optimal if and only if there

exists a power allocationQm1,1, ...,QmK ,K with∑K

k=1 tr(

Qmk,k)

= P such that

maxkµmkλmax

(

Hmk,kZ−1k HH

mk,k

)

≥ maxm,mk,k

λmax

(

[µm − µmk]+Hm,kHH

m,k +min(µm, µmk)Hm,kZ−1k HH

m,k

)

(5.7)

whereλmax(·) denotes the maximum eigenvalue andZk = I + HHmk,k

Qmk,kHmk,k.

Proof. First, since the objective is strictly monotonous inP the sum power budget has to be

exhausted. The remainder is based on the examination of the KKT conditions assuming that


the setm1, ...,mK is served. Consider the first KKT condition of usermk which is active on

subcarrierk by assumption. SettingQm,k = 0 for all m, mk it follows from (5.5) that

µmkHmk,kZ−1k HH

mk,k +Φmk,k = λ(P)I (5.8)

since the sum simplifies to∑M

s=mkcs = µmk. We writeλ(P) to denote that the Lagrangian multi-

plier is sum power dependent. With the singular value decompositionHmk,k = Umk,kΛ1/2mk,k

VHmk,k

(5.8) can be diagonalized leading to the set of equations

µmkλ(i)mk,k

1+ λ(i)mk,k

p(i)mk,k

+ φ(i)mk,k= λ(P) (5.9)

where·(i) denotes theith spatial channel. So for anyP > 0 we have

maxkµmkλmax

(

Hmk,kZ−1k HH

mk,k

)

= λ(P) (5.10)

sinceφ(1)mk,k= 0 for at least onek. On the other hand using (5.5) the first KKT condition for the

users not served on subcarrierk result in

max(µm − µmk, 0)Hm,kHHm,k +min(µmk, µm)Hm,kZ−1

k HHm,k +Φm,k = λ(P)I . (5.11)

The second addend occurs for all users while the first term is non-zero only for usersm < mk.

SinceΦm,k � 0 it follows

λ(P) = λmax

(

max(µm− µmk, 0)Hm,kHHm,k +min(µmk, µm)Hm,kZ−1

k HHm,k +Φm,k

)

∀m, k

≥ λmax

(

max(µm− µmk, 0)Hm,kHHm,k +min(µmk, µm)Hm,kZ−1

k HHm,k

)

∀m, k

≥ maxm,k

λmax

(

[µm − µmk]+Hm,kHH

m,k +min(µmk, µm)Hm,kZ−1k HH

m,k

)

. (5.12)

Combining (5.10) and (5.12) leads to (5.7) as a necessary condition. On the other hand (5.7) is

sufficient since then Lagrangian multipliers{Φm,k}M,km,k=1 can be found such that the KKT condi-

tions in (5.10) and (5.12) are fulfilled. Further the case of equality in (5.7) is well defined since

we can show thatλ(P) is a continuous monotonic function inP. This concludes the proof. �

Corollary 5.2. For sum capacity, i.e.µ1 = ... = µM, the above theorem simplifies to

maxkλmax

(

HHmk,kHmk,k

)

≥ maxm,k

λmax

(

HHm,kHm,k

)

∀m, mk. (5.13)

Note that the FDMA optimality condition in case of sum capacity in (5.13) is independent

of the SNR. This is in contrast to the general case.

Corollary 5.3. The optimality condition for single user transmission follows immediately from

(5.7) by setting m1 = m2 = ... = mK.


It is not very likely that FDMA is indeed the optimal transmitstrategy. As in the single

antenna case the probability decreases as the number of users or the number of subcarriers

increases. Moreover, an increasing number of antennas makes FDMA optimality less probable.

This is due to the fact that the decision on a subcarrier for a certain user with a strong spatial

eigenmode also implies a decision for his remaining spatialeigenmodes. In other words, the

probability that the strongest eigenmodes on one subcarrier belong to one single user decreases

as the number of antennas (and thus spatial modes) increases. Interestingly, the conditions

are very easy to check: Once the subcarrier assignment is fixed the optimization of the power

allocation is a simple linearly weighted water-filling procedure [12] or equivalentlymulti-level

water-filling as Hoo et. al termed it in the context of single antenna systems [78]. To find

the optimal FDMA subcarrier assignment nevertheless remains a combinatorial task having

exponential complexity.

5.1.4 Influence of the Number of Users on the Expected MIMO-OFDM

Sum Capacity

After FDMA optimality conditions for the general weighted rate sum problem and time-invariant

channels were introduced in the previous section we focus onthe long term throughput in the

following. The question of interest in this section is how the expected throughput scales with an

increasing number of users and which loss is implied by FDMA transmission of which we know

that it is suboptimal in general. Although not compliant with the traditional notion of ergodic

sum capacity a sum power constraint is imposed in each block fading state. This setting is more

meaningful from a technical point of view since the peak power is limited in practice. First

we investigate the influence in case of the optimum transmit strategy which implies nonlinear

precoding. Then we incorporate an FDMA constraint and characterize the asymptotic loss due

to this restriction.

Concerning the asymptotic behavior of frequency flat MIMO systems some results have

been published recently [100, 101]. In [100] the authors compared the optimal transmission

scheme to time division multiple access (TDMA) and characterized the asymptotic influence of

SNR and number of antennas. In [101] the dependence on the number of users was analyzed

for the optimum scheme and TDMA. However, this framework does not apply to the scenario

at hand due to the multiple parallel subcarriers. A further challenge is the correlation among

subcarriers due to oversampling.

System Model and Fading Statistics

First the assumptions made on the system model and the fadingstatistics are introduced which

are in particular important for the following results. The channel transfer function of userm on


subcarrierk is given by means of the FFT as

Hm,k =

L∑

l=1

Hm[l]e− j2π(l−1)(k−1)/K (5.14)

where [Hm[l]] s,t denotes the channel impulse response of thelth tap from transmit antennat

of the base station to thesth receive antenna of userm. Additionally we allow for correlation

at the transmit array and assume that the channel matrix of each tap can be decomposed to a

deterministic matrix reflecting the transmit correlation and a matrix with circular symmetric

complex Gaussian i.i.d. entries [Hw]s,t ∼ CN(0, 1/L) for all sandt

Hm[l] = HwRm[l]1/2 (5.15)

whereHw ∈ CnU×nB andRm[l] ∈ CnB×nB. With h(i)Tm being theith row of Hm[l] the transmit

correlation matrix is given by

Rm[l] = E{

h(i)m [l]h(i)

m [l]H}

.

Note thatRm[l] is independent of the receive antenna indexi. In [102] it is shown that under

the assumptions made aboveHm,k ∼ HwR1/2m with Rm =

∑L−1l=0 Rm[l]. We assumeRm to have full

rank for all usersm. In the following derivation we first assume thatRm = I nB. Later we show

that any positive definite transmit correlation matrix doesnot change the results.

Scaling of the MIMO-OFDM Sum Capacity

We define the expected sum capacity of a MIMO-OFDM BC according to an instantaneous

power constraintP in each fading stateH as

Csum(P) = E{

Csum(

P,H)}

(5.16)

with the instantaneous sum capacity given by the expressionin the dual MAC

Csum(

P,H)

= maxQm,k�0,

∑M,Km,k=1 tr(Qm,k)≤P

1K

K∑

k=1

log

∣∣∣∣∣∣∣

I nB +

M∑

m=1

HHm,kQm,kHm,k

∣∣∣∣∣∣∣

. (5.17)

The expressionCsum(P) will be called ergodic sum capacityin the following. Note that in

contrast to the traditional notion of ergodic capacity the expression (5.16) does not allow for an

adaptive power allocationamongfading states. In order to prove the main result of this Section

in Theorem5.8we need the following definition.

Definition 5.4. Let f (x) andg(x) be functions defined onR. We sayf (x) = O(g(x)) if and only

if

limx→∞

∣∣∣∣∣

f (x)g(x)

∣∣∣∣∣≤ c (5.18)


wherec is positive constant.

Further the subsequent bound on the expectation of a random variable will be of importance.

Lemma 5.5. Let X be a real random variable. The expectationE {|X|} is upper and lower

bounded by∞∑

n=1

Pr{|X| ≥ n} ≤ E {|X|} ≤ 1+∞∑

n=1

Pr{|X| ≥ n}. (5.19)

Proof. see [75, p.17] �

Another prerequisite is given by Lemma5.6. It characterizes the scaling behavior of the

maximum ofχ2 random variables scales. Note that the result takes into account correlation

among subcarriers of a specific user sincexm,k1 andxm,k2 are not assumed to be independent.

Lemma 5.6. Let Xm,k with 1 ≤ m ≤ M and 1 ≤ k ≤ K be identically chi-square distributed

random variables Xm,k ∼ χ2(2d) with 2d degrees of freedom. Further let Xm1,k1 and Xm2,k2 be

independently distributed for m1 , m2 and any k1, k2. Then

Pr{

λl ≤ maxm,k

Xm,k ≤ λu

}

= 1− O(log M)−1 (5.20)

with λl = log M + (d − 2) log logM andλu = log M + d log logM.

Proof. We have

Pr

{

maxm,k

Xm,k < λl

}

≤ Pr{

maxm

Xm,k < λl

}

= Pr{

Xm,k < λl}M

=

1− e−λl

d−1∑

k=0

1k!λk

l

M

≤(

1−C1e− f (λl )

)M

= exp(

M log(1−C1e− f (λl ))

)

. (5.21)

C1 is a positive constant independent ofλl and f (x)def= x− (d− 1) logx. Since log(1− x) ≤ −x,

this expression can be further bounded

exp(

M log(1−C1e− f (λl ))

)

≤ exp(

−MC1e− f (λl )

)

.


With λl = log M + (d − 2) log logM and abbreviatingβ = (d− 2) log logM we get

Pr

{

maxm,k

Xm,k < λl

}

≤ exp(

−C1e−β+(d−1) log(logM+β)

)

= exp(−C1 log M

)

=O(

1M

)

. (5.22)

On the other hand we have

Pr

{

maxm,k

Xm,k > λu

}

≤M,K∑

m,k=1

Pr{

Xm,k > λu}

= MKe−λu

d−1∑

k=0

1k!λk

u

≤ MKC2e− f (λu) (5.23)

with C2 a positive constant independent ofλu > 1. If we setλu = log M + d log logM and

abbreviateγ = d log logM we get the bound:

Pr

{

maxm,k

Xm,k > λu

}

≤ KC2e−γ+(d−1) log(logM+γ)

= KC2e− log logM

= O(

1log M

)

. (5.24)

Combining (5.22) and (5.24) leads to (5.20). �

Corollary 5.7. In a similar manner we can derive a lower bound onPr{

maxm,k Xm,k > λu}

:

Pr

{

maxm,k

Xm,k > λu

}

≥Pr{

maxm

Xm,k > λu

}

≥1−(

1−C2e− f (λu)

)M

≥1− exp(

−MC2e− f (λu)

)

=1− e−1

log M

≥ clog M

(5.25)

where the last inequality follows from1− exp(−1/x) ≥ c/x with c being a constant.

With the previous results at hand we are able to prove the mainresult which characterizes

the behavior ofCsum(P) for a large number of usersM.

Theorem 5.8. Assume no transmit correlation, i.e.Rm = I nB for all users m∈ M. Then the


achievable maximum sum-rate of the MIMO-OFDM BCC(P) given in(5.16) scales like

Csum(

P)

ϕ (M)= 1+ O

(

1log M

)

(5.26)

whereϕ (M)def= nB log logM.

Proof. The proof is split into two parts: First we will derive an upper bound on the sum capacity.

Subsequently, we find a lower bound.

The first part of the proof follows [100]. With the received signal on subcarrierk given by

yk = nk +∑M

m=1 HHm,kxm,k and the inequality det(A) ≤ (tr(A)/n)n, A ∈ Cn×n the sum capacity can

be upper bounded by

Csum(

P)

≤E

max∑M

m=1∑K

k=1 E{

xm,kxHm,k

}

≤P

1K

K∑

k=1

log

E{

yHk yk

}

nB

nB

=nB

K

K∑

k=1

E

log

En

{

nHk nk

}

+

M∑

m=1

Ex

{

xHm,kHm,kHH

m,kxm,k

}

− lognB

.

This expression can be further bounded

Csum(

P)

≤nB

K

K∑

k=1

E

{

log

(

1+P

KnBmax

m||Hm,k||2

)}

≤nB log

(

1+P

KnBE

{

maxm,k||Hm,k||2

})

(5.27)

where|| · || denotes the Frobenius-norm of a matrix and the last inequality follows from Jensen’s

inequality. The squared Frobenius norm of a matrixM ∈ CnB×nU with uncorrelated entries

[M ]s,t ∼ CN(0, 1) is chi-square distributed with 2nBnU degrees of freedom:

||M ||2 ∼ χ2(2nBnU).

Using the abbreviationhmax = maxm,k ||Hm,k||2 and withθ(M) = log M + nBnU log logM the

upper bound becomes

Csum(

P)

≤nB log

(

1+P

KnBE {hmax}

)

≤nB log(

1+P

KnB

[

θ(M) Pr{hmax≤ θ(M)}

+ E {hmax | hmax> θ(M)}︸︷︷︸

Z1(M)

Pr{hmax> θ(M)}︸︷︷︸

Z2(M)

])

≤nB log

(

1+P

KnB(θ(M) + Z1(M)Z2(M))

)

. (5.28)


Using Lemma5.5the termZ1(M) can be upper bounded as

Z1(M) = E {hmax | hmax> θ(M)}

≤ 1+∞∑

n=0

Pr{hmax> n | hmax> θ(M)}

= 1+ ⌊θ(M)⌋ +∞∑

⌈θ(M)⌉Pr{hmax> n | hmax> θ(M)}

= 1+ ⌊θ(M)⌋ +∞∑

⌈θ(M)⌉

Pr{hmax> n}Pr{hmax> θ(M)} .

A lower bound for the denominator of the last addend can be derived withθ(M) >> 1 and using

Corollary5.7

Pr{hmax> θ(M)} ≥ clog M

. (5.29)

So

Z1(M) ≤1+ ⌊θ(M)⌋ + log Mc

∞∑

⌈θ(M)⌉MKC1e

− f (θ(M))

≤1+ log M + nBnU log logM +MK log M

c× e− f (log M+nBnU log logM−1)

=1+ log M + nBnU log logM + O(1).

With Lemma5.6we get forZ2(M)

Z2(M) = Pr(hmax> θ(M)) = O(

1log M

)

.

With these expressions forZ1(M) andZ2(M) the upper bound in (5.28) becomes

Csum(

P)

≤ nB log

(

1+P

KnB

(

log M + nBnU log logM + O(1))

)

= nB log logM + O(

log logMlog M

)

(5.30)

which completes the proof of the upper bound.

A lower bound on (5.16) ca be obtained applying channel inversion

Csum

(

P)

≥ nB

K

K∑

k=1

E

{

log

(

1+P

KnB

(

maxm

λmin(Hm,kHHm,k)

))}

whereHm,k is anB × nB matrix. Note that channel inversion does not require any cooperation

among receive antennas. FornB < nU this matrix arises from deleting the lastnU − nB rows of


eachHm,k. For the more probable contrary casenB > nU , we add anynB − nU rows of another

users matrix on the same carrier. Hence, the resulting square matrixHm,k has i.i.d. Gaussian

entries and the number of independent matricesHm,k on each carrier is⌊nU/nBM⌋.Theorem 5.5 in [103] states that for any complex Wishart-MatrixM ∈ Cm×m = AA H, A ∈

Cm×m with i.i.d. as,t ∼ CN(0, 1) holds:

mλmin(M ) ∼ χ2(2)

SinceHm,kHHm,k is a Wishart-Matrix and with Lemma5.6we know that for largeM

maxm

λmin

(

Hm,kHHm,k

)

> 1.

Hence, channel inversion does not imply any ’power penalty’and in contrast brings along power

savings. Therefore the lower bound can withκ(M) = log nU

nBM − log log nU

nBM and λmink =

maxmλmin(Hm,kHHm,k) be expressed as

Csum(P) ≥ 1K

K∑

k=1

nBE

{

log

(

1+P

KnBλmink

)}

≥ 1K

K∑

k=1

nB log

(

1+P

Kn2B

κ(M)

)

Pr

{

λmink ≥1nBκ(M)

}

≥ 1K

K∑

k=1

nB log

(

1+P

Kn2B

κ(M)

)

1− O

1log nU

nBM

.

So we finally have the lower bound

Csum(P) ≥ nB log logM + O(

log logMlog M

)

. (5.31)

Combining (5.30) and (5.31) finally leads to (5.26) which is the desired result. �

With this result at hand it no problem to include additionally any arbitrary full rank transmit

correlation. The basic idea is to show that any full rank correlation matrix results in an additional

linear factor not influencing the asymptotic scaling. Note that this does not mean that there is

no difference in performance for any fixed number of users.

Corollary 5.9. Assume an arbitrary set of transmit correlationsRm ≻ 0 for all users m∈ M.

Then Theorem5.8still holds.

Proof. For the case of transmit correlation matricesRm ≻ 0 for all mwe get for the upper bound


in (5.27) with Hm,k = Hwm,kR1/2m

Csum(P) ≤nB log

(

1+P

KnBE

{

maxm,k||Hm,k||2

})

≤nB log

(

1+P

KnBE

{

maxm,k||Hwm,k ||2||R1/2

m ||2})

=nB log

(

1+PKE

{

maxm,k||Hwm,k ||2

})

. (5.32)

Hence the proof can be carried out as before. The following consideration confirms the lower

bound: Incorporating transmit correlation and denoting the minimum eigenvalue ofRm by

λmin(Rm), channel inversion now leads to

Csum(P) ≥ 1K

K∑

k=1

nBE

log

1+

PKnB

maxm

λmin

(

Hm,kHHm,k

)

λmin(Rm)

. (5.33)

SinceRm is full rank by definition for allm we haveλmin(Rm) > 0 and hence the transmit

correlation results in an additional positive factor not influencing the asymptotic scaling. �

Scaling of the MIMO-OFDM Sum Rate under FDMA Constraints

Theorem5.8 states that the MIMO-OFDM sum capacity basically scales asnB log logM, i.e.

double logarithmically with the number of users in the system. The linear factornB can be

interpreted as the number of spatial modes which the base station can serve in parallel. This

result fits well with the frequency flat case studied in [101]. The question is now whether an

additional FDMA constraint influences this scaling behavior.

The following theorem provides an answer to this question.

Theorem 5.10. Assume no transmit correlation, i.e.Rm = I nB for all m ∈ M. Then the

maximum achievable FDMA sum-rateCsumFDMA scales asymptotically as

limM→∞

CsumFDMA

(

P)

Csum(

P) = min

(

1,nU

nB

)

. (5.34)

Proof. Again, we derive an upper and a lower bound. The maximum achievable FDMA sum-

rateCsumFDMA(P) can be expressed as

CsumFDMA

(

P)

= E{

CsumFDMA

(

P,H)}

(5.35)


with

CsumFDMA

(

P,H)

= maxQm,k�0,

∑M,Km,k=1 tr(Qm,k)≤P

mk∈M ∀k

1K

K∑

k=1

log∣∣∣I + HH

mk,kQmk,kHmk,k

∣∣∣ .

(5.36)

With the singular value decompositionUm,kΣm,kVHm,k = Hm,k, Pm,k = diag(eig(Qm,k)), tr(Pm,k) =

tr(Qm,k) and the determinant identity∣∣∣I nU + HH H

∣∣∣ =

∣∣∣I nB + HHH

∣∣∣ this expression can be rewritten

as

CsumFDMA

(

P)

=E

maxPm,k�0,

∑M,Km,k=1 tr(Pm,k)≤P

mk∈M ∀k

1K

K∑

k=1

log∣∣∣I nmin + Σ

Hmk,kPmk,kΣmk,k

∣∣∣

≤ 1K

K∑

k=1

nmin log

(

1+P

KnBE

{

maxmk

||Hmk,k||2})

(5.37)

wherenmin = min(nB, nU). From here, the proof follows the proof of the upper bound inThe-

orem5.8 with nB replaced bynmin. The lower bound is also equivalent to the lower bound in

Theorem5.8 with the difference thatHm,k is of reduced dimensionality, i.e.Hm,k ∈ Cnmin×nmin.

This is obvious, since in a MIMO system the smallest number ofantennas limits the capacity

gain. Combining the two bounds states

limM→∞

CsumFDMA

(

P)

min(nB, nU) log logM= 1. (5.38)

and hence completes the proof. �

Corollary 5.11. Corollary 5.9carries over to the FDMA setting. In other words, the transmit

correlation matricesRm ≻ 0 for all m do not have any influence on the asymptotic scaling

behavior.

According to Theorem5.10the FDMA constraint does not have any influence on the double

logarithmic scaling with the number of users. Neverthelessthe linear factor depending on the

antenna constellation is now limited to the minimum of the number of transmit and receive

antennas. This is due to the fact that only min(nB, nU) spatial modes can be used under the

FDMA constraint. IfnB ≤ nU there is no difference between nonlinear precoding and FDMA

in the limit of many users. This is once again compliant with the results from [100] and [101].

In closing, it remains to be said that the presented results basically confirm the behavior known

from the frequency flat case for frequency selective scenarios.


5.2 Power Minimization

In the previous section it was shown that in case of weighted rate-sum maximization the polyma-

troid structure of the rate region for a fixed transmit strategy determines the optimum decoding

order. This important structural property generalizes from single antenna systems to multiple

antenna systems due to the fact that the MAC in both cases is characterized by arank function

as specified in (2.16) and (2.34). Applying TheoremA.8 then yields the optimum vertex and

thus decoding order. In this section now we turn towards Problem 2. As turns out in Section

5.2.1it is not straight forward to generalize the methodology developed in the single antenna

context for weighted sum power minimization under individual rate constraints.

5.2.1 Power Minimization under Individual Rate Constraints

In order to stay as general as possible we consider the weighted sum power minimization prob-

lem formulation taking place in the MAC as introduced in Section 3.3.1. The BC setting then

follows simply if the weights are chosen to be equal.

Recall the problem given in (3.21)

minM∑

m=1

λmPm subj. to R ∈ C(H,P1, ...,PM), (5.39)

whereλ ∈ RM+ is a vector of power weights. With (2.33) and (2.35) the capacity region

C(H,P1, ...,PM) is given by

C(H,P1, ...,PM) =⋃

Qm,k�0∑K

k=1 tr(Qm,k)≤Pm ∀∈M

R :

∑

m∈IRm ≤ f (I) ∀ I ⊆ M

.

where

f (I) =1K

K∑

k=1

log

∣∣∣∣∣∣∣

I +∑

m∈IHH

m,kQm,kHm,k

∣∣∣∣∣∣∣

. (5.40)

A simple reformulation of (5.39) then yields

minQm,k�0

M∑

m=1

λm

K∑

k=1

tr(

Qm,k)

subj. to1K

K∑

k=1

log

∣∣∣∣∣∣∣

I +∑

m∈IHH

m,kQm,kHm,k

∣∣∣∣∣∣∣

≥∑

m∈IRm for all I ⊆ {1, ...,M}

(5.41)

Since the objective is linear and the constraints are concave (5.41) is a convex problem and can

be solved by standard methods such as interior point or cutting plane methods. This fact was

observed first in [59]. Note that each of the constraints corresponds to a specificsubset of users

5.2. Power Minimization 101

and one constraining face of the polymatroid. Using the YALMIP software package [104], the

problem can be solved in a passable time for a moderate numberof degrees of freedom. The

parametersM = 3, K = 16, nB = 4 andnM = 4 worked well in our simulations. However,

the computational demand becomes prohibitive quickly since the number of constraints grows

exponentially with the number of users in the system (although at mostM of the constraints

can be active at the same time). Further this formulation does not take any advantage of the

structure of the capacity region. This is in contrast to the single antenna case studied in Section

3.3.1. There the key idea was to formulate the power minimization problem as an optimization

over contra-polymatroids. This was possible since the rankfunction characterizing the capacity

region on each subcarrier was generalized symmetric according to Definition A.6 from the

appendix. A simple application of TheoremA.9 yielded the optimum vertex on each subcarrier

rendering the problem convex. The reformulation in terms ofthe received powers and then in

terms of rates resulted in a reduction toM separable constraints allowing for an iterative water-

filling solution. Returning to MIMO systems, the functionf (I) in (5.40) is a submodular rank

function. However,f (I) is notgeneralized symmetricand thus LemmaA.7 does not apply. The

consequence is that a contra-polymatroid structure does not exist. This important property is

lost in the case of multi-antenna systems.

Recently an alternative approach avoiding the plain convexoptimization with 2M − 1 con-

straints was proposed in [59] and [105] independently. It was shown that dual optimization

methods can be used. The non-differentiable dual is optimized using the so-called ellipsoid

method which creates a sequence of shrinking ellipses containing the optimum Lagrangian

multipliers [63, 64]. while considering minimum rate requirements. The methodconsists of an

infinite sequence of weighted rate-sum maximizations. These can be solved with the algorithms

presented in Section5.1.2. This idea will prove useful in Section5.3. Another approach was

taken in [14]. There it was proposed to generalize Algorithm2 presented in Section3.2.3. An

iterative adjustment of each user’s Lagrangian multiplierwas used in order to meet the individ-

ual rate constraints. Although simulations suggest that the algorithm converges to the global

optimum it was not possible to prove the crucial monotonicity properties needed in Alg.2.

5.2.2 Sum Rate Iterative Water-Filling

In the following we consider the special case where only a sumrate constraintR instead of

individual rate constraints has to be met. This assumption simplifies the problem consider-

ably. Fairness among users may suffer in this case and strict QoS demands per user are not

supportable. However, there is a strong motivation for thisscenario: The resulting problem is

the complementary problem of classical downlink throughput maximization under a sum power

constraint, where a nice iterative water-filling algorithmexists [88]. Further this problem arises,

e.g., if a constant flow of information has to be guaranteed inorder to avoid a buffer overflow

but minimum power should be spent. The individual demands are not the relevant optimization


criteria, however, if the fading statistics are symmetric,i.e. the same for all users, even long

term fairness is provided. If we consider the dual uplink andadditionally introduce user specific

weights the problem also has an interpretation in the context of sensor networks. If many nodes

have to deliver redundant information to a common central node but their life-time limiting

factor is the remaining battery energy, it may make sense to weight each users transmit power

according to their current battery status in the cost function [17]. However this means that the

information to be transmitted is known to all nodes. A weighting also indicates that it is not

trivial to generalize the dual sum power iterative water-filling [88] to te present case.

With user specific power weightsλ ∈ RM+ we arrive at following problem statement:

minQm,k�0

M∑

m=1

λm

K∑

k=1

tr(

Qm,k)

subj. to1K

K∑

k=1

log

∣∣∣∣∣∣∣

I +m∑

m=1

HHm,kQm,kHm,k

∣∣∣∣∣∣∣

≥ R,

(5.42)

Note that (5.42) results from settingI = {1, ...,M} in (5.41) and omitting all other 2M − 2

constraints. Ifλ = [1, ..., 1]T the objective results in sum power. A geometric interpretation is

the following. We are looking for a polymatroid achievable with minimum weighted sum power

such that the sum capacity plane achieves the desired throughput. Note that if a solution to the

problem in (5.41) with individual rate constraints lies on the sum rate planeboth problems are

equivalent. We will further comment on this later. On the other hand, by varying the vector

λ ∈ RM+ , all Pareto-optimal power tuples achieving the desired throughput can be traced out

[17]. Obviously (5.42) is a convex optimization problem and can be solved with any standard

optimization method. However, we are interested in a smart and more efficient solution. Since

the notation is becoming quite complex anyway we consider the caseK = 1 in the following.

This is done without loss of generality and the derived results can be easily generalized toK

parallel subcarriers as the numerical examples will illustrate.

The Lagrangian of the considered problem is then given by

L(

{Qm}Mm=1, {Φm}Mm=1, µ)

=

M∑

m=1

λm tr (Qm) + µ

R− log

∣∣∣∣∣∣∣

I +M∑

m=1

HHmQmHm

∣∣∣∣∣∣∣

−M∑

m=1

tr (ΦmQm) ,

whereµ ∈ R+ is the Lagrangian multiplier for the sum rate constraint andthe matricesΦm � 0

are the Lagrangian multipliers corresponding to the positive semi-definiteness of the covariance


matrices. The Karush-Kuhn Tucker optimality conditions yield

∂L(

{Qm}Mm=1, {Φm}Mm=1, µ)

∂Qm=0 ∀m ∈ M (5.43)

R− log

∣∣∣∣∣∣∣

I +M∑

m=1

HHmQmHm

∣∣∣∣∣∣∣

=0

tr (ΦmQm) = 0 Φm � 0 Qm �0 ∀m ∈ M

The first KKT condition (5.43) can be rewritten as

1µ

(λm I −Φm) = Hm

I +M∑

n=1

HHn QnHn

−1

HHm ∀m ∈ M.

Unfortunately the rate constraint is not separable. All users are coupled by the common dual pa-

rameterµ. However, separability is a necessary condition for the application of block coordinate

ascend algorithms, where the iterative water-filling algorithm is a special case of [60].

An Equivalent Problem

Non-separable constraints occur in [88] in the context of throughput maximization under a sum

power constraint, too. To circumvent this difficulty the authors introduce an equivalent problem

with separable constraints.Equivalentin this context means that the optima of the original and

the equivalent problem are identical. We will try to adopt this approach in the following.

minSm,n�0

1M

M∑

n=1

M∑

m=1

λm tr(Sm,n)

subj. to1M

M∑

n=1

log

∣∣∣∣∣∣∣

I +M∑

m=1

HHmSm,nHm

∣∣∣∣∣∣∣

≥ R

(5.44)

Since the constraint is concave and the objective is affine, this problem is convex, too. Moreover,

both problems have the same solution:

Lemma 5.12.Let {Qm}Mm=1 be a set of optimizing transmit covariance matrices of the problem in

(5.42) and{Sm,n}Mm,n=1 a set of optimizing transmit covariance matrices of the problem in(5.44).

Then the objectives are equal at the optimum:

M∑

m=1

λm tr(

Qm

)

=

M∑

m=1

M∑

n=1

λm tr(

Sm,n

)

.

The proof follows that of [88, Thm. 1] exchanging the role of objective and constraint. For

the sake of completeness we carry out the proof.

Proof. Assume{Qm}Mm=1 is a set of optimizing transmit covariance matrices of the problem in


(5.42). Then we can choose the covariance matrices in (5.44) to be

Sm,n = Qm ∀m, n ∈ M

fulfilling the rate constraint and yielding the same sum power as in (5.42). Thus the optimum

of (5.42) is achievable in (5.44).

Now let {Sm,n}Mm,n=1 be a solution to (5.44). Setting

Qm =1M

M∑

k=1

Sm,n ∀m ∈ M

preserves the weighted sum power while meeting the rate constraint

log

∣∣∣∣∣∣∣

I +M∑

m=1

HHmQmHm

∣∣∣∣∣∣∣

≥ 1M

M∑

n=1

log

∣∣∣∣∣∣∣

I +M∑

m=1

HHmSm,nHm

∣∣∣∣∣∣∣

= R

due to the concavity of the log| · |-function. As a consequence each sum power achievable in

one problem is achievable in the other and vice versa. In particular the optima are identical.�

Due to Lemma5.12equivalently (5.44) can be solved. To this end we formulate the La-

grangian of (5.44)

L(

{Sm,n}Mm,n=1, {Φm,n}Mm,n=1, λ)

=1M

M∑

n=1

M∑

m=1

λm tr(

Sm,n)

+ µ

R−1M

M∑

n=1

log

∣∣∣∣∣∣∣

I +M∑

m=1

HHmSm,nHm

∣∣∣∣∣∣∣

−M∑

n=1

M∑

m=1

tr(

Φm,nSm,n)

.

The KKT conditions yield

∂L(

{Su,v}Mu,v=1, {Φu,v}Mu,v=1, λ)

∂Sm,n=0 ∀m, n ∈ M (5.45)

MR−M∑

n=1

log

∣∣∣∣∣∣∣

I +M∑

m=1

HHmSm,nHm

∣∣∣∣∣∣∣

=0 (5.46)

tr(

Φm,nSm,n)

= 0 Φm,n � 0 Sm,n �0 ∀m, n ∈ M

Again the first KKT condition in (5.45) can be rewritten as

∂LSm,n= λm I −Φm,n − µHm

I +M∑

u=1

HHu Su,nHu

−1

HHm = 0 ∀m, n ∈ M. (5.47)

The advantage of this alternative problem formulation is not obvious at first glance: There are

M2 optimization variables and all covariance matrices are still coupled according to the common


rate constraint. Hence the constraint is still not separable. This is in contrast to [88] where

this reformulation yields separable constraints allowingfor an iterative water-filling algorithm.

However, as will be shown subsequently (5.44) can be solved in an iterative manner avoiding

the problem arising from non-separability.

An Iterative Water-Filling Algorithm

In this section an iterative algorithm inspired by [88] is presented which solves the problem

in (5.44). The basic idea is to apply a modified version of a cyclic block coordinate descent

method with an additional averaging step exploiting the concavity of the objective. The key step

in order to develop an iterative algorithm is to define in which order the covariance matricesSm,n

in (5.44) are updated. To a bijective index mapping is introduced

πn :M→M , n ∈ M (5.48)

with the properties

πn(m) , πn(u) ∀ n ∈ M, m, u (5.49)

and

πn(m) , πu(m) ∀ m ∈ M, n , u. (5.50)

Note that the mapping defined by (5.48), (5.49) and (5.50) is not unique but the definition

assures thatM⋃

n=1

M⋃

m=1

(

m, πn(m)) ≡ M2,

i.e. every pair of indices occurs exactly once. Although theindexing seems rather complicated

at first glance it is needed to ensure that the right set of covariance matrices is updated. This

will become clear in the following. Applying the introducedindexing to the first KKT condition

(5.45) leads to

∂LSm,πn(m)

=(

λm I −Φm,πn(m)) − µHm

I +∑

t,u:πt(u)=πn(m)

HHu Sπt(u)Hu

−1

HHm ∀m, n ∈ M. (5.51)

The indexm specifies the user,n is a (cyclic) step index andπn(m) denotes the addend within

the relaxed sum rate constraint (5.44). Note that the sum in (5.51) is over all users and mappings

such that the addend they occur in is the same. According to (5.49) and (5.50) only covariance

matrices ofother users with adifferentcyclic step index occur in (5.51). We introduce a loop

over the indexn so that each cyclic iteration consists ofM steps. In each step the matrices

Sm,πn(m) for a fixed step indexn and for allm are updated. This optimization affectsM matrices,

i.e. one covariance matrix per user occurring in a different addend. To be more precise, in each


stepn = 1, ...,M of iterationi the following problem is solved:

{

S(i)m,πn(m)

}M

m=1= arg min

Sm�0

M∑

m=1

λm tr (Sm)

subj. to1M

M∑

m=1

log∣∣∣Zm,n + HH

mSmHm

∣∣∣ ≥ R

where the interference termZm,n is defined as

Zm,n = I +∑

l<n,s:πl (s)=πn(m)

HHs S(i)

s,πl (s)Hs+

∑

l>n,s:πl (s)=πn(m)

HHs S(i−1)

s,πl (s)Hs,

In order to solve (5.52) define the effective channels in stepn of iterationi as

G(i)m,n = HmZ−1/2

m,n . (5.52)

with the eigenvalue decompositionG(i)m,nG

(i)Hm,n = UmDmUH

m whereUm is unitary andDm diagonal

with positive entries. Note that the covariance matricesS(i)m,πl(m) occurring in the first addend are

known because they denote matrices which have been already updated in iterationi sincel < n.

Then the first KKT condition of (5.52) reads as

S(i)m,πn(m) = Um

[

µ

λmI − D−1

m

]+

UHm m= 1, ...,M. (5.53)

where [A]+ is defined component-wise.

Equation (5.53) has a water-filling structure although the dual parameterµ is unknown so

far. Note that the individual weightsλm result in nothing but a scaling of the water-filling level.

Taking the KKT conditions for afixed step indexn, water-filling can be performed overM

matrices each belonging to a different user and occurring in a different addend of the constraint

(5.46). The difficulty is to chooseµ such that the rate requirementR is met. By doing so the

feasible set is not left at any time. Since the sum rate is independent of the decoding order the

rate already achieved by the remaining covariance matricescan be subtracted

R(i)w f(n) = max

MR−M∑

m=1

log∣∣∣Zm,n

∣∣∣ , 0

(5.54)

andµ is chosen such thatR(i)w f(n) is met. Any one-dimensional search algorithm can be used

to determine the water-filling level, since the achieved rate is monotone in the dual parameter

µ. Alternatively rate water-filling over the parallel spatial channels can be used avoiding the

one-dimensional search.

Although the described procedure fulfills stepwiseM out of M2 KKT-conditions, it does

not necessarily constitute a convergent sequence. The sequence of dual parametersµ (i.e. the


water-filling levels) may oscillate and thus not jointly fulfill the entire set of KKT-conditions.

However, using the concavity of the log| · |-function on the set of positive definite matrices

convergence can be established: To this end after an entire cycle a convex combination of the

covariance matrices has to be taken in order to avoid that thealgorithm gets stuck. This leads

to a provably convergent process described in Alg.9. Since the fixed sum rate determines the

water-filling level the algorithm is calledsum rate iterative water-filling.

Algorithm 9 Sum Rate Iterative Water-Filling

initialize covariance matricesS(0)m,n = 0nU for all m, n

while desired accuracy is not reacheddofor n = 1 to M do

(1) calculate effective channelsG(i)m,n according to5.52for all usersm

(2) determineS(i)m,πn(m) by water-filling according to (5.53) whereµ is adjusted such that

R(i)w f (n) in (5.54) is met

end for(3) averageS(i)

m,πn(m) ← 1M

∑Mn=1 S(i)

m,πn(m) for all m, n ∈ Mend while

Lemma 5.13.Algorithm9 converges a set of covariance matrices achieving the globaloptimum

of (5.44).

Proof. Each iteration of the algorithm consists of an inner loop with M water-filling steps and

a subsequent averaging step. Denote the set of covariance matrices resulting from thenth step

in iterationi as

S(i)n =

{

{S(i)m,π1(m)}, ..., {S

(i)m,πn(m)}, {S

(i−1)m,πn+1(m)}, ..., {S

(i−1)m,πM(m)}

}

.

and the set of covariance matrices after the averaging at theend of theith iteration as

S(i) ={

S(i)m,πn(m)

}M

m,n=1.

Define the constraint function

g(

S(i)n

)

=1M

M∑

u=1

log

∣∣∣∣∣∣∣∣∣∣

I +∑

s,l≤n:πl (s)=u

HHs S(i)

s,πl (s)Hs+

∑

s,l>n:πl (s)=u

HHs S(i−1)

s,πl (s)Hs

∣∣∣∣∣∣∣∣∣∣

and the objective function

f(

S(i)n

)

=∑

u≤n

M∑

m=1

tr(

S(i)m,πu(m)

)

+∑

i>n

M∑

m=1

tr(

S(i−1)m,πu(m)

)

.

In each stepn of the inner loop the algorithm finds the unique optimum covariance matrices

{S(i)m,πn(m)} given all other covariances such that the power is minimizedand the rate constraint is


fulfilled:

{S(i)m,πn(m)}

Mm=1 = arg min

S:Sm�0

M∑

m=1

λm tr (Sm)

subj. to1M

M∑

m=1

log∣∣∣Zm,n + HH

mSmHm

∣∣∣ ≥ R

(5.55)

where the interference termZm,n is defined as

Zm,n = I +∑

l<n,s:πl (s)=πn(m)

HHs S(i)

s,πl (s)Hs+

∑

l>n,s:πl (s)=πn(m)

HHs S(i−1)

s,πl (s)Hs.

The definition of each step implies

g(S(i)n ) ≥ R ∀n

and combined with the concavity of the log| · |-function

g(

S(i+1))

≥ g(

S(i)M

)

≥ R

it can be seen that the sequenceS(i) remains within the feasible set. FurtherS(i) is defined on a

compact set (since it can be bounded by single user water-filling solutions) and thus has at least

one limiting pointS. The definition of each step in (5.55) establishes a monotone sequence

f (S(i)) = f (S(i)M) ≥ ... ≥ f (S(i)

1 ) ≥ f (S(i−1))

which is bounded from below. This implies that the sequence converges to a limit

limi→∞

f (S(i)) = f (S).

It now has to be shown thatS minimizes f (S). To this end consider a subsequence{S(in)|n =1, 2, ...} converging toS. It is sufficient to show that the sequence{S(in+1)} converges to the same

limiting point S [60, pp. 273-274]. Since the water-filling solution of each stepis unique we

have:

limi→∞S(in) = lim

i→∞S(in+1)

1 = S. (5.56)

Due to (5.56) the covariance matrices{Sm,π1(m)}Mm=1 are water-filling solutions with respect to

{Sm,πn(m)}Mm=1,n,1. Further, in the limitS all matrices of each userm are equal

Sm,π1(m) = ... = Sm,πM(m) for all m

and thus all covariances are jointly water-filling solutions, i.e. the KKT conditions are jointly


fulfilled. Hence it follows

limi→∞

f (S(i)) = f (S).

�

Remark 5.14. An important observation to speed up the algorithm is that after the first water-

filling step within the first iteration the rate constraint isalready met. This is due to the fact that

the rate to be achieved in the first step isR(1)w f (1) = MRaccording to (5.54). Hence the following

M − 1 water-filling steps within the first iteration can be omitted.

Algorithmic Improvements

The crucial operation in Alg.9 is the averaging of covariance matrices in step 3. Note that

this is in contrast to the sum power iterative water-filling in [88] where the concavity is not

exploited explicitly for updating the covariance matricesin the original version of the algorithm.

However, due to the concavity of the log| · |-function averaging the sets of covariance matrices

increases the sum rate at any time. Motivated by this fact Alg. 9 can be simplified and improved.

This is already indicated by Remark5.14. Leaving out the subsequentM − 1 iterations is

equivalent to an averaging after the first iteration. A consequent improvement is not to wait one

entire loop, i.e.M water-filling steps but to perform an averaging after every single step. Since

in this caseM − 1 covariance matrices of each user are necessarily equal, only one matrix per

user has to be stored. Thus the necessary memory reduces fromM2 to M covariance matrices

[88]. The update simplifies to

S(i)m ←

M − 1M

S(i−1)m +

1M

S(i)m ∀m ∈ M.

whereS(i)m is the covariance of userm in iteration i. Even more general the update can be

described as

S(i)m ← (1− ν)S(i−1)

m + νS(i)m ∀m ∈ M, ν ∈ (0, 1].

Since the entire loop consists only of one update and all other matrices are equal the expressions

for the effective channels in (5.52) and the rate to be allocated (5.54) simplify to

G(i)m = Hm

I +∑

l,m

HHl S(i−1)

l H l

−1/2

(5.57)

and

R(i)w f = max

MR−M∑

n=1

log

∣∣∣∣∣∣∣∣∣∣

I +M∑

l=1ł,n

HHl S(i−1)

l H l

∣∣∣∣∣∣∣∣∣∣

, 0

(5.58)

Although only the choiceν = 1/M is motivated by the equivalence of the modified problem


in (5.44) and the original problem in (5.42), an arising question is the following: Are there other

possible choices forν such that the generated sequence converges to the global optimum? The

following Lemma answers this question.

Algorithm 10 Generalized Sum Rate Iterative Water-Filling

initialize covariance matricesS(0)m = 0 for all m

while desired accuracy is not reacheddo(1) calculate effective channelsG(i)

m according to (5.57) for all m(2) determineS(i)

m by doing water-filling such thatR(i)

w f in (5.58) is met

(3) updateS(i)m according to

S(i)m ← (1− ν)S(i−1)

m + νS(i)m ∀m ∈ M

where in first iterationi = 1 setν = 1/M,and in subsequent iterationsi > 1:

opt. 1)

ν = ν0 ∈(

0,1M

]

or opt. 2)

ν = max θ

subj. to log

∣∣∣∣∣∣∣

I +M∑

m=1

HHm

(

(1− θ) S(i−1)m + θS(i)

m

)

Hm

∣∣∣∣∣∣∣

≥ R

end while

Lemma 5.15.Let ν0 be a fixed update factor for Algorithm10with

0 < ν0 ≤1M.

Then Algorithm10converges to the global optimum of(5.42).

Proof. The proof is identical to the proof of Lemma5.13. The only difference is how to show

that the feasible set is not left and the constraint is met forall n.

To this end define the constraint function

g(

S(i))

= log

∣∣∣∣∣∣∣

I +M∑

m=1

HHmS(i)

m Hm

∣∣∣∣∣∣∣

where

S(i) = {S(i)1 , ...,S

(i)M}. (5.59)


Further define the sets

S(i)m = {S

(i−1)1 , ...,S(i−1)

m−1 , S(i)m ,S

(i−1)m+1 , ...,S

(i−1)M } (5.60)

and

S(i) = {S(i)1 , ..., S

(i)M}.

Then it can be shown by induction that

g(S(i)) ≥ R for all i.

Set i = 1. Iteration 1 withν = 1/M assures that the algorithm starts with a power allocation

within the feasible set

g(

S(1))

= g

1M

M∑

m=1

S(1)m

≥1M

M∑

m=1

g(

S(1)m

)

= R (5.61)

sinceg (·) is concave on the set of positive semidefinite matrices. The addition and division of

sets (5.61) is defined as the addition and division of its ordered elements. Now consider iteration

i + 1. It holds

g

1M

M∑

m=1

S(i+1)m

≥1M

M∑

m=1

g(

S(i+1)m

)

= R

and thus

g

α1M

M∑

m=1

S(i+1)m + (1− α)S(i)

≥ R (5.62)

due to the concavity ofg (·) for all α ∈ [0, 1]. Substituting the definitions (5.59) and (5.60) in

(5.62) leads to

g(

S(i+1))

= g(

ν0S(i+1) + (1− ν0)S(i))

≥ R for all ν0 ∈[

0,1M

]

whereν0 = 0 is trivially excluded. This concludes the proof. �

The use of a fixed step-sizeν0 ∈ (0, 1/M] is denoted in Alg.10 asoption 1. Figure5.1

illustrates the influence ofν0 on the convergence. On the left side the case withν0 ≤ 1/M is

depicted. As stated in Lemma5.15the algorithm converges but the convergence speed decreases

asν0 decreases. The right side of5.1 shows the caseν0 ≥ 1/M for the same setting as in the

left. Obviously the algorithm does not converge any more although the initial progress is larger.

However, the fact that in each iteration the decrease of the objective depends linearly on

ν motivates a second modification of Alg.10 indicated byoption 2. Consequently in each

iterationi the factorν should be chosen as large as possible without violating the rate constraint


10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5

3

3.5

4x 10

4

Iterations

sum

pow

erν

0= 0.1

ν0= 0.06

ν0= 0.04

10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5

3

3.5

4x 10

4

Iterations

sum

pow

er

ν0 = 0.1

ν0= 0.2

ν0= 0.3

Figure 5.1: Convergence behavior of Algorithm10 for MIMO BC with λ = [1, ..., 1]T, nB = 4,nM = 4, M = 10 users andR = 60 bps/Hz. Convergent scenarios with different valuesof ν0 ≤ 1/M (left) and divergent scenarios withν0 ≥ 1/M (right). The convergence speeddecreases asν0 decreases

R in order not to leave the feasible set:

ν = max θ

subj. to log

∣∣∣∣∣∣∣

I +M∑

m=1

HHm

(

(1− θ) S(i−1)m + θS(i)

m

)

H

∣∣∣∣∣∣∣

≥ R

By this means in each iteration the update factor is chosen such that a convex combination

of covariance matrices achieves the rate constraint with minimum sum power. The proof of

Lemma5.15simply extends to this case. In the context of (weighted) rate-sum maximization a

similiar step-size optimization has been used in [106, 91]. Due to the concavity of the constraint

the optimization can be done by a simple bisection search. Needless to say, in order to avoid

extensive evaluations of the objective function any approximate solution can be used. This is

illustrated in Fig.5.2exemplarily. The tiny red crosses mark the evaluations during the bisection

process. Note thatR(ν) is concave but not necessarily monotonously decreasing inν as can be

seen in on the right of Fig.5.2. Thus even iterations withν = 1 may occur causing a maximum

possible decrease of the objective.

Note that the use of the secure choiceν = 1/M in the first iteration of Alg. 10 is not

necessary. It simplifies the convergence proof, however, itcan be shown that for 0< ν0 ≤ 1/M

the algorithm converges even without choosingν = 1/M in iteration i = 1. An advantage of

the initial common iteration is that it improves comparability of the algorithms. In general Alg.

10 using the optimized step-size indicated asoption 2converges much faster than with fixed

step-size corresponding tooption 1. This is illustrated by the upper part of Figure5.3 which

compares the convergence behavior forM = 10 andM = 100 users withnB = 4 andnM = 4

antennas, equal weightsλ = [1, ..., 1]T and a sum rate requirement ofR = 40bps/Hz. It can be


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 58.2

58.4

58.6

58.8

59

59.2

59.4

59.6

59.8

60

60.2

ν

Sum

rat

e

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 59.8

59.85

59.9

59.95

60

60.05

ν

Sum

rat

e

Figure 5.2: Optimization ofν as indicated byoption 2in Alg. 10 for an exemplary system withM = 10 users. Two exemplary steps withνopt = 0.137 (left) andνopt = 0.575 (right) for thesame setting as in Fig.5.1

observed that the initial version (Alg.9) as well as option 1 of Alg.10suffer from an increasing

number of users. This is due to the fact the influence of each elementary update is limited to

1/M thus slowing down the convergence speed. This behavior is analog to that observed in [88]

in the context of sum rate maximization. In contrast the number of users seems to have hardly

any influence on the convergence speed for the step-size optimized version of Alg.10. Note

that this comes at the cost of additional evaluations of the objective function during the one-

dimensional step-size optimization. Nevertheless even for M = 100 users and as much as four

antennas at the base station as well as at the user equipmentsthe algorithm converges within

approximately 5 steps.

Up to now all examples considered the frequency flat case. As mentioned the considered

algorithms also apply to systems withK subcarriers. The lower part of Fig.5.3 considers the

a scenario withM = 10 users andK = 256 random subcarriers. Comparing with the upper

left plot it can be seen that the number of parallel channels does not influence the speed of con-

vergence. However, the computational demand grows linearly with the number of subcarriers

K since all operations (such as the calculation of effective channels) have to be carried outK

times.

Sum Rate Iterative Water-Filling and Individual Rate Constraints

The iterative water-filling algorithms presented in the previous section can be applied in case

that only a sum rate constraint is imposed. Unfortunately itis not straightforward to generalize

the procedure to the case of individual rate constraints studied in Section5.2.1. However, the

presented algorithms can be used to provide a lower bound on the necessary sum power. Let

R ∈ RM be the vector of individual rate constraints and denote the sum rate to be achieved by

Rsum=∑M

m=1 Rm. Then Algorithms9 and10 can determine the sum power necessary to achieve


10 20 30 40 50 60 70 80 90 1000

200

400

600

800

1000

1200

Iterations

sum

pow

er

Alg. 9

Alg. 10 (step−size 1/M)

Alg. 10 (optimized step−size)

50 100 150 200 250 300 350 400 450 5000

200

400

600

800

1000

1200

Iterations

sum

pow

er

Alg. 9Alg. 10 (step−size 1/M)Alg. 10 (optimized step−size)

10 20 30 40 50 60 70 80 90 1000

200

400

600

800

1000

1200

1400

Iterations

sum

pow

er

Alg. 9Alg. 10 (step−size 1/M)Alg. 10 (optimized step−size)

Figure 5.3: Comparison of convergence speed for exemplary random channel withnB = 4,nU = 4, andR = 40 bps/Hz for M = 10 users andK = 1 subcarrier (top left),M = 100 usersandK = 1 subcarrier (top right) andM = 10 users andK = 256 subcarriers (bottom)

Rsum. This gives a lower bound on the power necessary to achieve the individual rates since the

optimization takes place over an enhanced set. Comparing (5.42) to (5.41) only a single out of

2M −1 constraints remains. However, if at the optimum of (5.41) only the sum rate constraint is

active the solution of (5.41) coincides with the solution of (5.42). In other words, if the desired

rate tuple lies on the sum rate plane of the optimum polymatroid the presented algorithms can

be used to solve the problem with individual rate constraints. This is illustrated in Figure5.4.

In all other cases the solution provides a lower bound on the necessary sum power.

5.3 Minimum Rate Requirements

With the insights from the preceeding sections we turn towards Problem3 incorporating mini-

mum rate constraintsR ∈ RM+ . As the results from weighted rate sum-maximization in Section

5.1 suggest the problem is more complicated than in the single antenna setting. Recall that

there we reformulated the problem in terms of the rates per user and subcarrier and applied a

5.3. Minimum Rate Requirements 115

012345

0 1 2 3 4 5

0

1

2

3

4

5

R1

R2

R3

Figure 5.4: Polymatroid corresponding to the optimal transmit covariances for MIMO MACwith M = 3 users, 2× 2 random channels and desired rate vectorR = [3 3 3] bps/Hz lying onthe sum rate plane

simple block descent algorithm interpretable as rate water-filling (see Alg.5). The fact that the

optimum decoding order is known on each elementary subcarrier allowed to explicitly state the

problem in convex form (in fact the objective was log-convex).

Let us conjure up the problem statement of Problem3 from Section3.1:

max µTR subj. to R ∈ C(H, P)

R ≥ R(5.63)

A reformulation in terms of rates allowing an algorithmic solution in the single antenna case

is not possible any more since no contra-polymatroid structure exists. Nevertheless, in analogy

to (3.17) we can define the feasible set as the intersection of half-spaces with the convex capacity

region yielding a convex set:

R f (H, R, P) ={

R ∈ RM+ : R ∈ C(H, P),R ≥ R

}

(5.64)

Lemma3.4obviously applies also in the present multi-antenna case since only the convexity of

the capacity region is exploited: Using (5.64) we can rewrite the problem as

maxR∈R f (H,R,P)

M∑

m=1

µmRm. (5.65)

Since (5.63) consists of the maximization of an affine function over a convex set it a convex

problem. Unfortunately it is not clear how to solve (5.63) algorithmically since the formulation

is only implicit.


Optimization in the Dual Domain

Similar to [59, 105] the crucial idea is to solve5.63in the dual domain. To this end consider

the Lagrangian of5.63

L(R, µ′) =M∑

m=1

µmRm+

M∑

m=1

µ′m(Rm− Rm).

We assume that the feasible set is non-empty withR ∈ C(H, P − ǫ) whereǫ > 0. By this

means the Mangasarian-Fromovitz constraint qualificationis fulfilled via Slater’s Theorem so

that Lagrangian multipliers exist and the duality gap is zero [107, 60]. Further we keep the sum

power constraint implicit in the following. Maximizing with respect to the primal variables (i.e.

R) yields the dual function

g(µ′) = supR∈C(H,P)

L(R, µ′)

where the supremum can be replaced by the maximum sinceC(H, P) is compact. Now define

coefficients

µm = µm+ µ′m for all m= {1, ...,M} (5.66)

summing up the Lagrangian multipliersµ′ and the initial weightsµ. Then the dual function can

be rewritten as

g(µ) = maxR∈C(H,P)

M∑

m=1

µmRm−M∑

m=1

µ′mRm. (5.67)

Obviously (5.67) is an affine version of (5.1) which can be efficiently solved by the algorithms

discussed in Section5.1.2. Furtherg(µ) is Lipschitz-continuous onRM+ . However, to minimize

the dual function may be complicated since differentiability with respect toµ is not guaranteed.

Thus optimization methods which do not rely on derivatives or gradients have to be used. In

general any cutting plane method is suitable for minimizing(5.67).

In the following we make use of the ellipsoid method which wasoriginally introduced by

Khachiyan [108] and has been applied successfully several times for related problems recently

[59, 105]. The ellipsoid method generates a sequence of shrinking ellipses containing the solu-

tion. It can be interpreted as a smart kind of cutting plane method which rules out half spaces

according to the evaluation of a subgradient∗ and then covers the remaining set by a new ellipse

with minimum volume. In Figure5.5the ellipsoid method is illustrated for a two user example.

At an arbitrary pointµ a subgradient can be found using the definition of the dual function: Let

R be the solution to

R = arg maxR∈C(H,P)

L(R, µ).

∗For further details the interested reader is referred to [64, 63].


Then

g(µ) = maxR∈C(H,P)

L(R, µ)

≥ L(R, µ)

= g(µ) +M∑

m=1

µm − µm)(Rm− Rm).

(5.68)

Defining the vector

ν = R − R (5.69)

it can be easily seen using (5.68) that

ν ∈ ∂g(µ),

where∂g(µ) is the subdifferential ofg(µ) at µ. Thus for a givenµ the half space of dual

parameters corresponding to

µ : (µ − µ)Tν ≥ 0

can be ruled out. The entire procedure is summarized in Alg.11. In the following subsection we

broach the issue of algorithm initialization which is not trivial due to the possibly very limited

size of the feasible set. Moreover we comment on the fifth stepof Alg. 11.

Bounds on the Optimum Lagrangian Multipliers µ∗ for Algorithm Initialization

In order to apply the ellipsoid or any other cutting plane method a necessity is that an initial set

containing the solution can be specified. To this end upper bounds on the Lagrangian multipliers

are needed. In many cases this is not a major problem since a feasible point may be easy to find.

Unlike in [59] where the search can be limited to the unit sphere it is not trivial to find an

initial ellipsoid covering the optimal Lagrangian multipliersµ∗ in the first step of Alg.11. The

following lemma provides a bound on the optimum dual variablesµ∗:

Lemma 5.16.Let

µ∗ = arg minµ∈RM

+

g(µ)

be the optimum dual variables, with g(µ) defined in(5.67). Assume that there exists a rate tuple

R ∈ relint(

R f (H, R, P))

whererelint (·) denotes the relative interior. Then

0 ≤ µ∗m ≤ θm for m withRm > 0 (5.70)

where

θm =

∑Mn=1 µn

(

Rsun (P) − Rn

)

Rm− Rm(5.71)


Algorithm 11 MIMO-OFDM minimum rates algorithm

(1) initialize µ(0) according to

µ(0)m =

θm/2 if Rm > 0

0 otherwise

with θm defined in (5.71) and choose an initial ellipseΓ(0) with centroidµ(0) such that

||Γ(0)1/2(x − µ(0))|| ≤ 1

for all x with 0 ≤ xm ≤ θm andm= 1, ...,Mwhile desired accuracy not reacheddo

(2) with µ(n) defined in (5.66) solve

R(n) = arg maxC(H,P)

M∑

m=1

µ(n)m Rm

(3) determine subgradientν(n) according to

ν(n) = R(n) − R

(4) update ellipse

Γ(n+1) =

M2 − 1M2

Γ(n) +

2M − 1

ν(n)ν(n)T

ν(n)TΓ(n)−1

ν(n)

with new centroid

µ(n+1) = µ(n) +1

M + 1Γ

(n)−1ν(n)

√ν(n)TΓ(n)−1

ν(n)

(5) assure thatµ(n+1) ∈ RM+

end while

with Rsun (P) being the maximum single user rate corresponding to the water-filling solution

Rsun (P) = max

Qk�0:∑K

k=1 tr(Qk)≤P

1K

K∑

k=1

log∣∣∣I +HH

n,kQkHn,k

∣∣∣ . (5.72)

Proof. An upper bound ong(µ∗) can be found as follows.

g(µ∗) =L(R∗, µ∗)

=

M∑

n=1

µnR∗n +

M∑

n=1

µ∗n(R∗n − Rn) (5.73)

≤M∑

n=1

µnRsun

The last inequality follows from the fact that the second addend in (5.73) is zero andRsun ≥ R∗n

for all n.


On the other hand we have

g(µ∗) =L(R∗, µ∗)

≥L(R, µ∗)

=

M∑

n=1

µnRn +

M∑

n=1

µ∗n(Rn − Rn) (5.74)

≥M∑

n=1

µnRn + µ∗m(Rm − Rm)

Combining (5.73) and (5.74) and solving forµ∗m leads to (5.70). �

Note that (5.72) is very easy to solve andRsun (P) can be calculated using water-filling over

the inverse eigenvalues ofHHn,kHn,k for all k. In principle any upper bound on

∑Mn=1 µnR∗n can be

used: For example

M∑

n=1

µnR∗n ≤ max

nµn

M∑

n=1

R∗n

≤ maxnµnRsum

(5.75)

can be used whereRsum is the sum capacity:

Rsum= maxQ:

∑Mn=1

∑Kk=1 tr(Qn,k)≤P

1K

K∑

k=1

log

∣∣∣∣∣∣∣

I +M∑

n=1

HHn,kQn,kHn,k

∣∣∣∣∣∣∣

.

However, using the single user water-filling solutionsRsu is very appealing due to its simplicity.

It remains to find an interior point within the relative interior of the feasible setR ∈relint

(

R f (H, R, P))

. Defining th rate vector

Rǫ = [R1 + ǫ, ..., RM + ǫ]T

this can be done solving

Pǫ = min P

subj. to Rǫ ∈ C(H,P)(5.76)

while choosingǫ such thatPǫ < P. Note that the volume of the initial ellipse and thus the

number of iterations necessary to achieve a certain accuracy in Alg. 11depends on the tightness

of the derived bounds onµ∗.


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

µ1

µ2

µ(0)

µ(1)

µ(2)

µ∗

Figure 5.5: Exemplary convergence behavior of Algorithm11for a random channel withM = 2users andµ∗ = [0.4 0]

Convergence Issues

Lemma 5.17.The sequence of centroidsµ(n) generated by Algorithm11 converges to the opti-

mum Lagrangian multipliers.

limn→∞

µ(n) = µ∗. (5.77)

Proof. The proof is standard. See e.g. [63]. �

It remains to say some words about step 5 of Alg.11 which is not specified in detail up

to now. It is not clear from the beginning which rate constraints will turn out to be active or

not (see e.g. user 3 in Figure5.6and user 2 in Figure5.5). If a rate constraintRm is not active

we consequently getµ∗m = 0. The dual (5.67) is defined onµ ∈ RM+ . However, the ellipsoid

algorithm may generate iteratesµ(n) with negative componentsµ(n)m < 0 such thatg(µ(n)) does

not exist. In this case choose a subgradientν with

νm =

−1 if µ(n)m < 0

0 otherwise(5.78)

and proceed with step 4 of the algorithm. This procedure is repeated untilµ(n+1) ∈ RM+ and

thus the negative half-space is ruled out. Note that no evaluation of the dual is needed while

shrinking the ellipse further. However, this may occur various times since in contrast to pure

cutting plane methods the ellipsoid method adds new parts tothe remaining feasible set (see

Figure5.5).

A further speed-up can be achieved if only the Lagrangian multipliers of users with strictly

positiverate requirements are considered since all other constraints are inactive and thus their


multipliers are zero anyway. For this purpose define the set

Mp ={

m ∈ M : Rm > 0}

. (5.79)

Then in the first step of Alg.11 only the multipliersµm of usersm ∈ Mp have to be initial-

ized and in step (4)M can be substituted by the number of users with positive requirements

|Mp|. As a result the problem dimension reduces fromM to |Mp| speeding up the convergence

significantly especially if only a view users have positive rate constraints.

10 20 30 40 50 60 70 801

1.5

2

2.5

3

3.5

Iterations

R [b

ps/H

z]

user 1

user 2

user 3

10 20 30 40 50 60 70 80

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Iterations

µ

user 1

user 2

user 3

Figure 5.6: Simulation example for a three user system withnB = 3, nU = 1, µ =[1/3 1/3 1/3]T and rate constraintsR = [3 2.5 0.5]T bps/Hz

Figure5.6 shows the convergence of Alg.11 for an exemplary three user system and a

random channel withK = 1. The base station hasnB = 3 antennas while the users ownnU = 1.

The weights are given byµ = [1/3 1/3 1/3]T (thus the objective is throughput) and the QoS


constraints areR = [3 2.5 0.5]T bps/Hz. The transmit SNR is 10dB. The upper plot depicts the

achieved rates and the lower plot the corresponding sum of weights and Lagrangian multipliers

µ = µ+ µ′. It can be observed that the QoS constraint of user 3 is not active resulting inµ∗3 = 0,

while users 1 and 2 achieve their rate requirements of 3 and 2.5 bps/Hz having active constraints

µ∗i > 0, i = 1, 2.

Optimum DPC Order

In Section3.4we discussed the optimum decoding orders for the problem at hand in the context

of single antenna systems. The optimum decoding order was determined on each subcarrier

by the ordering of the channel gains. On the other hand the polymatroid structure of the MAC

region over all subcarriers allowed to choose a global decoding order applicable on all subcarri-

ers. Transfered to the broadcast scenario this meant that superposition coding with an individual

decoding order on each subcarrier as well as dirty paper coding with a global precoding order

could be used to achieve the desired rate tuple.

In the present MIMO case the former option vanishes due to thenon-degradedness of the

vector broadcast channel while the polymatroid structure of the MAC remains. To be more

precise from (5.67) we know that for the optimumR∗, µ∗ we have

R∗ = arg maxR∈C(H,P)

M∑

m=1

µmRm. (5.80)

whereµ = µ + µ∗ andR∗ is the vertex of a polymatroid corresponding to the permutation

π(·) : µπ(1) + µ∗π(1) ≥ ... ≥ µπ(M) + µ

∗π(M). (5.81)

With Definition A.3 and TheoremA.8 it is thus optimal to decode userπ(M) first followed

by userπ(M − 1) and so on. This translates directly to the broadcast scenario using the reverse

order encoding userπ(1) first followed by userπ(2) and so on. So the optimum encoding order is

rather aresultthan a part of the formulation for Problem3 which is in contrast to pure weighted

rate sum-maximization from Section5.1.

Figure5.7 depicts rates andµ + µ∗ over the transmit SNR for an exemplary system with

random channel realizationsK = 1, nB = 3, nU = 1, µ = [0.3 0.2 0.5]T andR = [3 2 1]T .

Below a transmit SNR of∼ 11.35 dB the problem is infeasible since the rate requirement can not

be fulfilled. With increasing SNR the Lagrangian multipliers decrease monotonously. Similar

to the single antenna case the optimum decoding order changes over SNR as the ordering of the

elements ofµ changes. As the constraints become inactive the corresponding users’ rate starts

to increase. Note that there exist two fractions (users 2,3 and 1,3) within the SNR range where

µm = µn which corresponds to time-sharing among usersmandn.

5.4. Comments on FDMA Constrained Transmission 123

8 10 12 14 16 18 20 22 240

1

2

3

4

5

6

7

8

9

10

transmit SNR [dB]

R [b

ps/H

z]

user 1user 2user 3

infeasible

8 10 12 14 16 18 20 22 240

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

transmit SNR [dB]

µ

user 1user 2user 3

infeasible

Figure 5.7: Rate and weights over transmit SNR for an exemplary three user system withnB = 3,nU = 1, µ = [0.3 0.2 0.5]T and minimum rate requirementsR = [3 2 1]T bps/Hz

5.4 Comments on FDMA Constrained Transmission

In the current chapter we focused on the three reference problems in the context of multiple

antennas up to now. It was shown that all problems can be stated in convex form. Nevertheless

the corresponding transmission strategies are quite demanding and involve non-linear precoding

in the BC and successive decoding in the MAC. Thus it is interesting to consider all problems

under additional FDMA constraints as done for the case of a single antenna in Section4.3.3.

We briefly comment on this topic in the following.

In principle the framework presented in Section4.3.3 can be generalized to the case of

multiple antennas. In analogy to (4.61) we can introduce

GFDMA(H)def=

{

(R,P) ∈ RM+1+ : R ∈ CFDMA(H,P)

}

, (5.82)


where the set of all rate tuples achievable with sum powerP under an FDMA constraint is given

by

CFDMA(H,P)def=

⋃

M∑

m=1νm,k≤1 k∈K ,νm,k∈{0,1}

⋃

∑

m,kpm,k≤P

⋃

Qm,k:tr(Qm,k)≤pm,k

R ∈ RM

+ : Rm ≤1K

K∑

k=1

νm,k log∣∣∣I + HH

m,kQm,kHm,k

∣∣∣

.

As in the single antenna case the set (5.82) is not a convex set. However the expressions

(R1(µ, λ), ...,RM(µ, λ),P(µ, λ)) = arg maxGFDMA(H)

M∑

m=1

µmRm− λP. (5.83)

can be evaluated easily for anyµ ∈ RM+ , λ ∈ R+. The basic properties from Lemma4.14nor-

malization, monotonicityandunboundednessobviously carry over the scenario with multiple

antennas. An adaptive modification of the multipliersµ ∈ RM+ for this reason can be used to

meet demanded rates as in Alg.7.

More general, for any set of multipliersµ ∈ RM+ , λ ∈ R+ (5.83) yields a point on the convex

hull of GFDMA(H) with supporting hyperplane normal vector [µ − λ]T . Further solving

maxGFDMA(H)

M∑

m=1

µmRm− λP (5.84)

in (5.83) corresponds to the evaluation of the Lagrangian dual. For this reason the proposed

framework is suited to find lower bounds for all three reference problems under additional

FDMA constraints. The monotonicity properties guarantee that rate constraints as well as power

constraints can be met. Geometrically speaking this means that one moves on the convex hull

of GFDMA(H) by modifying the normal vector while taking into account the rate or power con-

straints reflected by hyperplanes orthogonal to the axes. This interpretation also explains why

the solution is suboptimal for non-convex problems: The convex hull obviously yields only an

outer bound on the setGFDMA(H). To summarize, if one is restricted to FDMA transmission,

dual approaches offer an interesting perspective for MIMO system optimizationunder FDMA

constraints which should be investigated more in detail.

5.5 Summary

In this chapter we studied resource allocation problems fortime-invariant MIMO-OFDM broad-

cast and multiple-access channels. The three reference problems presented in Section3.1in the

context of single antenna systems served as a guideline. In Section 5.1 we first focused on

weighed rate sum-maximization. We derived necessary and sufficient conditions for optimality

5.5. Summary 125

of exclusive subcarrier assignment being improbable in general. Subsequently, we focused on

the simpler case of sum capacity and studied the asymptotic influence of the number of users

in the system. We then turned towards the power minimizationproblem in Section5.2. In

this reagard a major difficulty was that the contra-polymatroid structure, which is important to

solve the power minimization problem for single antenna systems, is lost. The reason is that the

rank-function defining the polymatroid structure in the MIMO case is not generalized symmet-

ric. Nevertheless, the problem can be solved using an optimization approach taking place in the

Lagrangian dual domain. Since the algorithmic solution is quite demanding, we then focused

on the special case of a throughput constraint and derived anefficient algorithm based on rate

water-filling suitable for the optimization of large systems. We then returned to weighted rate

sum-maximization and considered additional minimum rate requirements in Section5.3. As

in the case of sum power minimization, the optimum decoding order is not known in advance.

Nevertheless, it is again possible to solve this problem using a dual approach based on the sub-

gradient, although the appealing water-filling algorithmsintroduced for the single antenna case

do not apply any more. Finally, in Section5.4we commented on how to generalize the dual op-

timization methods, which were introduced in the single antenna setting in order to incorporate

additional FDMA constraints, to multiple antenna senarios.

Chapter 6

Conclusions

In this thesis we studied the uplink and downlink of cellularOFDM systems from an information-

theoretic perspective. A special emphasis was put on rate requirements in order to reflect the

increasing importance of strict Quality of Service-constraints in real systems. The derived re-

sults allow to allocate resources under the considered constraints in a capacity-achieving man-

ner. While we first focused on systems equipped with a single antenna, the study was extended

to multiple-antenna systems afterwards.

At first we analyzed the single antenna OFDM up- and downlink for time-invariant channels.

We considered three reference problems in order to reflect different settings occurring in system

optimization for real systems. Two of these are weighted rate-sum maximization under a power

constraint and weighted transmit power minimization underindividual rate constraints. In order

to bring together the two conflicting approaches, we extended the first problem statement and

included additionally individual rate constraints. All problems are relevant for downlink as well

as uplink scenarios. For the broadcast scenario, we extended the results from [45] and solved

the third problem including minimum rate requirements. We then chose the uplink to address

the three reference problems. This perspective is different from the broadcast formulation and

yields various new insights. The algorithmic solutions have an appealing rate water-filling in-

terpretation while all problems are closely connected. We showed that there exists an optimum

successive decoding order which can be applied to all subcarriers in both settings. While one

would expect this for the multiple-access channel, this result is quite surprising for the broad-

cast scenario. There, the channel gains on each subcarrier determine the decodability of other

users’ codewords if Superposition Coding is applied. Despite this fact the optimum allocation is

always such that the same decoding order can be applied on allsubcarriers. This is of particular

interest, since in practice coding is often performed over frequency, which does not allow to

apply different decoding orders on different subcarriers.

Subsequently, we left the area of time-invariant channels and focused on delay-limited trans-

mission over single antenna OFDM broadcast channels using the metric of delay-limited capac-

ity introduced in [66]. The delay-limited capacity reflects the need for a constant mutual infor-

mation independent of the current fading state, allowing totransmit at the desired rate without

127

128 Chapter 6. Conclusions

paying attention to the dynamic change of the channel quality caused by the fading process.

In order to get insights into the general behavior of the delay-limited capacity, we investigated

the low and high SNR regime of point-to-point OFDM links. It was shown that the minimum

energy per transmitted bit, at which reliable communication is possible under delay constraints,

behaves as the reciprocal expected maximum channel gain andthus decreases as the number

of independent path gains increases. We characterized under which conditions on the fading

distribution delay-limited transmission is possible making more precise the notion of regular

fading distributions introduced in [67] for OFDM systems. Then we turned towards the multi-

user case and presented an algorithm to evaluate the delay-limited capacity region of OFDM

broadcast channels. Due to its practical relevance we subsequently studied power minimiza-

tion for time-invariant channels under rate constraints and the restricting assumption that each

subcarrier is assigned exclusively to a single user. The algorithmic solution for this non-convex

problem generalizes the results from the first part. Although generally the exclusive assignment

of subcarriers is suboptimal, the performance loss is surprisingly small even in case of discrete

power allocations as the number of subcarriers becomes large. For any resource allocation ful-

filling the rate constraints, the power loss can be further bounded using an expression based on

the duality gap. The results indicate that even with strict rate requirements exclusive subcarrier

assignment is a valuable alternative to optimum transmission avoiding non-linear operations in

practical single antenna systems.

Finally we extended our investigation of optimum transmit strategies for time-invariant

channels to systems with multiple antennas. Clearly, a major problem is the non-degradedness

of the elementary multiple-antenna broadcast channel on each subcarrier and the fact that the

contra-polymatroid structure in the multiple-access channel is lost. In contrast to the single

antenna case the decomposition into a set of parallel channels therefore does not simplify the

problems significantly. A reformulation in terms of rates isnot possible. This fact causes

the algorithmic solutions to become much more involved, especially in case of individual rate

constraints. Appealing algorithms exist only in special cases such as a common throughput

constraint. The optimum decoding and precoding orders result from the ordering of the Lan-

grangian multipliers and thus are not known in advance. An exclusive subcarrier assignment

avoids these difficulties. However, it comes at the expense of a possibly significant loss in

system performance, since not all spatial dimensions can beexploited if the base station owns

more antennas than the user equipments. Consequently, suboptimal and potentially linear trans-

mit strategies exploiting the spatial degrees of freedom seem to be a suitable choice for real

systems, especially if the channel varies quickly. A further interesting step towards practice

would be to relax the strict delay constraint of a single block in this thesis to a finite number

of blocks, so that the demanded rate is guaranteed in averageover this period of time. This

does not only correspond to situations occurring in practical systems but it also would provide

a challenging link to scheduling theory.

Appendix

Rank Functions, Polymatroids and

Contra-polymatroids

In order to characterize the capacity region of multiple-access channels the concept ofpolyma-

troids turns out to be elementary. Thus we introduce polymatroids,contra-polymatroids and

some related basic results.

Definition A.1. Let f : P(M) 7→ R+ be a set function mapping from the power setP(M) over

M = {1, ...,M} to the set of nonnegative real numbers. Thenf is called a submodular rank

functionif it satisfies the following three properties:

1. (normalized) f (∅) = 0

2. (nondecreasing)f (A) ≤ f (B) forA ⊂ B and

3. (submodular) f (A) + f (B) ≥ f (A∪B) + f (A∩B).

With DefinitionA.1 at hand polymatroids can be defined.

Definition A.2. Let f : P(M) 7→ R+ be a set function. The set

C( f )def=

x ∈ RM

+ :∑

n∈Sxn ≤ f (S), for all S ⊆ M

(A.1)

is called apolymatroidif f is a submodular rank function.

A polymatroid is thus a special polytope. It is a subset of thenonnegative orthant, limited

by 2M − 1 affine constraints. There exists a second equivalent definitionfor a polymatroid.

Definition A.3. Let f : P(M) 7→ R+ be a normalized set function and define for any permuta-

129

130 Chapter 6. Conclusions

tion π(·) onM a vectorx(π) ∈ RM+ with

xπ(M)(π) = f ({π(1), ..., π(M)}) − f ({π(1), ..., π(M − 1)})...

xπ(m)(π) = f ({π(1), ..., π(m)}) − f ({π(1), ..., π(m− 1)}) (A.2)...

xπ(1)(π) = f ({π(1)}).

Then the setC( f ) is called a polymatroid ifx(π) ∈ C( f ) for all M! permutationsπ(·).

The equivalence of these two definitions is well established[109, 110, 111]. In fact the

vectorsx(π) ∈ RM+ constitute theverticesof the polymatroid. Each corner point lying strictly

inside the positive orthant is a vertex corresponding to oneout of theM! permutations.

There is an analogon to DefinitionA.1 for supermodularity.

Definition A.4. Let f : P(M) 7→ R+ be a set function mapping from the power setP(M) over

M = {1, ...,M} to the set of nonnegative real numbers. Thenf is called a supermodular rank

functionif it satisfies the following three properties:

1. (normalized) f (∅) = 0

2. (nondecreasing)f (A) ≤ f (B) forA ⊂ B and

3. (supermodular) f (A) + f (B) ≤ f (A∪B) + f (A∩B).

Definition A.5. Let f : P(M) 7→ R+ be a set function. The set

D( f )def=

x ∈ RM

+ :∑

n∈Sxn ≥ f (S), for all S ⊆ M

(A.3)

is called acontra-polymatroidif f is a supermodular rank function.

A contra-polymatroid is thus also a subset of the positive orthant havingM! vertices. Loosely

speaking a contra-polymatroid can be thought of as a polymatroid mirrored at any hyperplane

with normal vectorn = 1M with 1M being the all ones vector of dimensionM.

To establish an important relation between polymatroids and contra-polymatroids we need

the following subset of the submodular rank functions from Definition A.1 calledgeneralized

symmetricrank functions.

Definition A.6. Let f : P(M) 7→ R+ be a submodular rank function andC( f ) be a polymatroid.

The rank functionf is calledgeneralized symmetricif there exists an increasing functiong and

a vectory ∈ RM+ so that

f (S) = g

∑

n∈Syn

for all S ⊆ M

131

Lemma A.7 ([57]). Let g be an increasing concave function and for each vectory ∈ RM+ let

fy(S) = g(∑

n∈S yn) be a generalized symmetric rank function. Then for any vector x ∈ RM+ the

set

D def=

{

y ∈ RM+ : x ∈ C( fy)

}

(A.4)

is a contra-polymatroid.

Proof. Assumeg is strictly increasing. Then the polymatroid is given by

x ∈ RM

+ :∑

n∈Sxn ≤ g

∑

n∈Syn

for all S ⊆ M

(A.5)

Thus we have for ally ∈ RM+ such thatx ∈ C( fy)

D =

y ∈ RM

+ :∑

n∈Syn ≥ g−1

∑

n∈Sxn

for all S ⊆ M

(A.6)

where fx = g−1 (∑

n∈S xn)

is obviously a normalized, nondecreasing and supermodularrank

function. ThusD is a contra-polymatroid. The generalizing case withg being non-decreasing

follows from basic limit and continuity arguments. �

The optimization of linear functions over (contra-)polymatroids simplifies due to the fol-

lowing important theorems.

Theorem A.8. Consider the linear optimization problem

maxM∑

m=1

µmxm subj. to x ∈ C( f ) (A.7)

whereµ ∈ RM+ , µ1 ≥ ... ≥ µM andC( f ) is a polymatroid. Then any vertexx(π) given in Definition

A.3with π = {1, ...,M} is an optimum of(A.7).

As one would guess an analog result exists for contra-polymatroids:

Theorem A.9. Consider the linear optimization problem

minM∑

m=1

λmxm subj. to x ∈ D( f ) (A.8)

whereλ ∈ RM+ , λ1 ≥ ... ≥ λM andD( f ) is a contra-polymatroid. Then any vertexx(π) given in

DefinitionA.3with π = {1, ...,M} is an optimum of(A.8).

For a proof see e.g. [109, 111]. Thus the optimal vertices are determined by the linear

factors.

Publication List

[1] T. Michel and G. Wunder, “Solution to the sum power minimization problem under given

rate requirements for the OFDM multiple access channel,” inProc. Annual Allerton Conf.

on Commun., Control and Computing, Monticello, USA, Sept. 2005.

[2] ——, “Minimum rates scheduling for OFDM broadcast channels,” in Proc. IEEE Int.

Conf. on Acoustics, Speech, and Signal Proc. (ICASSP), Toulouse, May 2006.

[3] T. Michel, C. Zhou, and G. Wunder, “A generic OFDM downlink scheduling policy incor-

porating service induced rate constraints,” inProc. of 15th IST Mobile& Wireless Com-

munications Summit, June 2006.

[4] G. Wunder and T. Michel, “Optimal resource allocation for parallel Gaussian broadcast

channels: minimum rate constraints and sum power minimization,” IEEE Trans. Inform.

Theory, vol. 53, no. 12, pp. 4817–4822, Dec. 2007.

[5] G. Wunder, T. Michel, and C. Zhou, “A framework for resource allocation in OFDM

broadcast systems,” inProc. European Signal Processing Conf. (EUSIPCO), Florence,

Sept. 2006.

[6] G. Wunder and T. Michel, “Delay-limited OFDM broadcast capacity region and impact of

system parameters,” inProc. IEEE Int. Information Theory Workshop (ITW), Mar. 2006.

[7] ——, “Approaching the delay-limited OFDM broadcast capacity region with OFDMA,”

in Proc. IEEE Workshop on Signal Processing Advances in Wireless Communications

(SPAWC), July 2006.

[8] ——, “Delay-limited capacity of OFDM broadcast channels,” eprint ArXiv, Oct. 2006,

available at:http://arxiv.org/pdf/cs/0610061.

[9] ——, “The low and high SNR behavior of the OFDM delay limited capacity,” inProc.

Annual Allerton Conf. on Commun., Control and Computing, Monticello, USA, Sept. 2006.

[10] ——, “On the OFDM multiuser downlink capacity,” inProc. IEEE Vehicular Techn. Conf.

(VTC), Los Angeles, Sept. 2004.

133

http://arxiv.org/pdf/cs/0610061

134 PUBLICATION LIST

[11] ——, “Multiuser OFDMA optimization: Algorithms and duality gap analysis,” inProc.

ITG/IEEE Int. Workshop on Smart Antennas (WSA), Darmstadt, Mar. 2008.

[12] T. Michel and G. Wunder, “Optimal and low complexity suboptimal transmission schemes

for MIMO-OFDM broadcast channels,” inProc. IEEE Int. Conf. on Communications

(ICC), Seoul, May 2005.

[13] G. Wunder and T. Michel, “A (not so) many user sum capacity analysis of the MIMO-

OFDM broadcast channel,” inProc. ITG/IEEE Int. Workshop on Smart Antennas (WSA),

Apr. 2005.

[14] ——, “Minimum rates scheduling for MIMO-OFDM broadcastsystems,” inProc. IEEE

Int. Symp. on Spread Spectrum Techniques and Applications (ISSSTA), Manaus, Aug.

2006.

[15] T. Michel and G. Wunder, “Sum rate iterative water-filling for Gaussian MIMO broad-

cast channels,” inProc. Intern. Symp. On Wireless Personal Multimedia Communications

(WPMC), San Diego, Sept. 2006.

[16] ——, “Achieving QoS and efficiency in the MIMO downlink with limited power,” inProc.

ITG/IEEE Int. Workshop on Smart Antennas (WSA), Vienna, Feb. 2007.

[17] ——, “Throughput aware power balancing for the MIMO Gaussian MAC,” in Proc.

IEEE Workshop on Signal Processing Advances in Wireless Communications (SPAWC),

Helsinki, June 2007.

[18] T. Haustein, S. Schiffermuller, V. Jungnickel, M. Schellmann, T. Michel, and G. Wunder,

“Interpolation and Noise Reduction in MIMO-OFDM - A Complexity Driven Perspec-

tive,” in ISSPA, Sydney, Aug./Sept. 2005.

[19] G. Wunder, C. Zhou, and T. Michel, “On minimum delay and buffer size - bounds for

stable OFDM broadcast systems,” inProc. IEEE Int. Symp. Information Theory and its

Applications (ISITA), Oct. 2006.

[20] C. Zhou, G. Wunder, and T. Michel, “Utility optimization over discrete sets,” inProc.

IEEE Int. Conf. on Communications (ICC), Glasgow, June 2007.

[21] T. Michel and G. Wunder, “Nonlinear downlink beamforming under QoS constraints: Op-

timum precoding order and the need for time-sharing,” inProc. IEEE Int. Symp. on Per-

sonal, Indoor and Mobile Radio Communications (PIMRC), Athens, Sept. 2007.

[22] G. Wunder, I. Blau, and T. Michel, “Utility optimization based on MSE for parallel broad-

cast channels: The square root law,” inProc. Annual Allerton Conf. on Commun., Control

and Computing, Monticello, USA, Sept. 2007.

PUBLICATION LIST 135

[23] G. Wunder and T. Michel, “On optimization of multiuser systems using interference cal-

culus,” inProc. Asilomar Conf. on Signals, Systems and Computers, Monterey, Nov. 2007.

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